Homework - a reflection and some questions to ponder

It's been a wee while....
In typical fashion, just as a few people tell me not to stop writing, I stop writing.  It's been a busy couple of weeks under the beam of the projector (doesn't have the same ring to it "as at the chalk face", does it?)

Anyway, thanks for some positive feedback about the ramblings so far. It's been longer than I'd hoped since I wrote my last little piece.  I felt that a break would be a good idea.  I didn't want to make this a place where I could come on and moan about "the bloody fourth year".  I wanted to get to deeper issues than that.  I felt last time round I maybe let the practicalities of the job get in the way of my higher order thinking.

I'm going to continue from where I left off last time, although this time I'll hopefully deal with the topic in a more reflective manner.  The issue I've been considering is homework.  I make no apologies for using my own practice as a start point in all of this.

Homework
I have an S4 credit class and an S6 set doing higher (for those in England - this means that although they are doing calculus etc and some of them will master it - it's going to be like pulling teeth for me to get them through it.)

First of all the S4.  The reason I was so frustrated in my last post was that I had designed all of my homework in S3 in a certain way.  It was designed to be consolidation of the current topic but it also contained questions which referred back to previous topics in order to keep them in the pupils memory.  My thinking is that by constantly going back to things from the past it will refresh pupils memory and hopefully become something that the pupil knows well.  This is why initially I was depressed with their early S4 homework when they couldn't do anything from last year!  Now I realise the error of my judgement.  The fact was that pupils could do things from early in S3 (further back in time!) because they'd had more consolidation of these topics through the homework.  Whereas the things from later in term (closer to the present time) they had forgotten as they hadn't had as much consolidation over the long term.  I am seeing some very positive results from this now.

My questions to you, readers, are the following?
a.) How do you structure Standard Grade/GCSE homework?
b.)  How do you reinforce previously learned concepts with a class who could do it at the time, but like my class, are unlikely to revise very much?
c.)  Do you use notes jotters?  I give my pupils notes.  The pupils all agree they like them, however I despair when I see questions which are almost identical to those in the notes left with no attempt!
d.)  How do you mark homework?  I have occasionally used peer marking and assessment for learning strategies, however I'm maybe a wee bit old fashioned in that I like to mark it and give them a score with some feedback points attached.  This is time consuming though, as we all know.  A colleague of mine never gives raw scores.  She runs with a green, amber, red system.  I think this may be a wee bit too wishy-washy for my preferences.

My fear is that I am trying to not only teach the kids the work, but also remember it for them, by tailoring all of my exercises as I do.  I don't think I give them enough responsibility for their own learning.  However, there are results to justify at the end of the day.  It's a delicate balance.

The "Success Loop"
 Moving on to the higher class.  I run a similar sort of system for this class.  However, I have additionally added two "recommendations" that some of them are following up.  These are that pupils at least read over their notes every day after class - preferably writing them up - this is how I learned at uni and during my own higher - although I know it might not be suitable for everyone. The second thing I have decided to ask pupils to do is consider the "success loop".

It looks a wee bit like below....

 I am not forcing pupils to do corrections, they need to have some motivation for themselves by the age of 17!  However, they should be considering the above. Most of the pupils in fairness have been trying to fix their mistakes now, rather than getting into dangerous mindsets like I did as a pupil "Oooh, I can never do those hard trig equations!".  My retort to the kids now is "Learn to do it then!".  I will help any of them over and over until they can do it, so it's not getting help that they have to worry about.

Another small, but not insignificant breakthrough that I have made with this class is by giving them a questionnaire about their own attitude to the study of the subject.  I have given questionnaires before about my own teaching - however I've never properly challenged the pupils to be introspective about their attitudes until now.  The impact during that subsequent two weeks has been great - three or four of the pupils who are borderline pass/fails really seem to have taken the message on board and are working harder now.  I feel this was much more effective than the usual rant.  I did add into the rant the fact that Uni is going to be so much harder and much more work etc and that they had to get used to it, and that if they couldn't deal with it maybe they should choose some other life path.   If you'd like  a copy of the questionnaire for a nosey drop me a line.

Any feedback, tips, disagreements etc - leave me a comment! :)

Maybe all this philosophy is pointless...

I've just sat and marked 30 S4 credit jotters.  The stuff I taught at the board old fashioned style, chalk and talk, with lots of examples, well, they have either managed this fine or completely forgotten it and not consulted their notes.  The stuff we used the group work, self-directed learning tasks, investigations etc., has been completed dreadfully without exception.

Why can it be?  I felt that the higher order thinking that was going on during some of these less conventional lessons was excellent.  I felt the kids were coming up with some great ideas.  Is it because we didn't do enough consolidation of the ideas once they had been discovered that the resultant homework is so bad?

Why won't the kids look up their notes from S3 on simultaneous equations?  We did zillions of it last year and they could do it inside out!  Looking at the homework some have done it fine but a lot didn't even know where to start.

I imagine most of you will have been in the position where you suffer 3 hours of hell where you feel yourself leaning harder and harder with the pen every time you see -4 x -1 = -4 !!!  Sometimes it's just laziness or carelessness on the kids part and sometimes it's just they don't know it very well.  Sometimes - like this evening, I can't figure out why it's all gone so badly.  However, I sure as hell know how to fix it - it will involve lots of chalk and talk etc, but surely...

*seeks inspiration from....*

A Mathematical Vision...

Prompted by feedback from various people I thought I’d post up some more of my thoughts.  As I said a couple of weeks back when I started this blog, I had intended it to be something that would help me to reflect upon what I do.  I am questioning whether it is helping me to focus my reflective thoughts more successfully than I had managed before.   One thing that is for sure is that some of the feedback has really got me thinking - lots of thoughts about the big, deep issues of mathematics teaching.  I didn’t really know what I was searching for when I set out, but now I think I realise.  I want to be able to define my pedagogy and principles more clearly.  I want to have a clearer idea of who I am as a teacher, and whether what I do is as effective or worthwhile as I believe it to be?

I feel I am getting closer to my own teaching philosophy - it‘s amazing the revelations I‘ve had in the space of a few weeks.  I know I work in a results driven business - just like football managers.  But I’d like to think I’m more of an Arsene Wenger - trust the kids and get them doing some beautiful maths moves rather than a Sam Allardyce - rigid, boring but gets results.  Ok Arsene, might not always win the trophies, but I don’t think good exam results and true mathematical enlightenment should be mutually exclusive.

As you can no doubt tell, I am no great writer so in order to stop this turning into a completely unstructured rant I am going to use points made by other people to write about some more of the things I have been thinking about over the past week or so.

I want my students to experience maths rather than being 'taught' by me. However am I asking these kids to reinvent the wheel?   I also worry that perhaps we need to teach them a set of tools and then give them problems where they have to decide which tools to use.

I think there must be an element of re-inventing the wheel with any learning process.  If we were to withdraw this aspect  I think we would end up with purely a “here’s a result. Use it to do questions 1 to 30” approach to teaching.  If kids can’t reason what point is there in equipping them with tools they can’t use?  I can teach anybody to differentiate, but what good is that if they can only apply it to a few selected styles of problems they have learned off by heart?  What happens if something unusual appears in a question in the exam?  PANIC!!!  All kids need a set of tools to use to solve problems, I agree with this.  However, I also feel that we shouldn’t just hand kids these tools without having at least tried to get them to understand them.  I remember as a pupil at school - I wasn’t interested in the proofs our teacher would do as they weren’t examinable.  Well, what a shock I got at uni once proof became important - I was completely useless at it, never having had to properly reason before in my life.

For instance if I face a new maths problem I have a degree and PhD from which I can draw knowledge to help me solve a particular problem. What do the kids have to help them! 

The kids have lots of knowledge - we should be gently building upon things they already know.  A problem is something the kids haven’t got the answer to yet - but as long as the problems set are appropriate - they will almost certainly have the maths to tackle it.  Generally in a lesson there are only a couple of new facts, and as a rule at least one of these could be figured out by the kids as it is a follow on to something they already know!

Surely we should be equipping them to open up new areas of knowledge.

Definitely, but without having ever managed to open up an area of maths for themselves which is already well understood by others, how can they ever move forward to the cutting edge of our subject with confidence and more importantly - the ability and experience to do so?  For example getting kids to figure out simple results by themselves that are part of the course lets them practice the reasoning and logic skills that would be needed should they ever go on to be adults working on new areas of research.  If a pupil has never experienced trying to prove something and not succeeding at the first attempt but still going back to revise his ideas - if he has never had this experience - what hope is there of him ever opening up new areas of the subject later on?

People learn maths by thinking. 

Yes.   But I question whether lots of repeated applications of the same formula is “thinking”.  I loved the anology about a dancer learning to dance by dancing.  I think our kids should be solving problems to get good at maths - but I mean proper problems, not boring repetition.  Although I don’t dispute that in some cases I agree that lots of practice is required to deal with subtle algebraic differences between questions etc.

The only reason to have all of the V,A,K learning styles is to get their attention levels up by having some variety. The V,A,K learning styles are the medium. But the medium is not the message. 

I really liked this post.  Thanks for that input.  I suppose I’ve been guilty of trying to overly jazz up lessons to tick these boxes.

Random points:
I think we need to have confidence in our kids. Today with my new first year class I had them doing an investigation into interior and exterior angels of polygons. My ultimate aim was a formula for the sum of the interior/exterior angles of a n-sided polygon.  I gave them very little information other than what an interior and exterior angle was. I wasn't very prescriptive at all and didn't give them a lot of input into what to do. However, despite this most of them followed the same steps. Most of them chose a table to organise their data for example. These kids had a plethora of skills for solving these problems already, the only new thing that was required was a wee piece of knowledge which as the teacher I gave them.  After that I set them free to discover things for themselves.  I’m a great believer in the idea that if you discover something yourself you are more likely to remember it.

Sometimes we should maybe focus on how much of our "chalk and talk time" is actually delivering properly new content. As I said in my previous post, why should I stand at the board and do examples on adding fractions with x+ 3 and x + 5 as the denominators, when they already know how to add fractions. Why not challenge them first? Push them. All I would do at the board is be providing stabilisers for them. Which does seem a wee bit untrusting, as the kids already have all of the algebra and fractional skills to do the problems. They just haven't put it together before! Let them tackle it in pairs. Let them articulate ideas, let them come up with their own solutions

I know that sometimes when having these debates it’s easy to be idyllic.  When confronted by a class of 30 hormonal 15 year olds, it is something altogether different.  But I think we sometimes need to be brave and design lessons in a way so that pupils can explore for themselves, while at the same time, ensuring that if they don’t quite get there - they have the knowledge required for the exam.

I'll finish on a quote from Larkin’s rant - he asked how much has changed in the years since it was originally written?

“I was made to learn by heart: ‘The square of the sum of two
numbers is equal to the sum of their squares increased by twice
their product.’ I had not the vaguest idea what this meant and
when I could not remember the words, my tutor threw the book at
my head, which did not stimulate my intellect in any way.”


As usual, please post your own thoughts and reflections etc - and tell me if you think I'm wrong.

Pre-conceptions challenged: Letting go of chalk and talk

After writing in my last post about being reluctant to let go of chalk-and-talk style teaching with certificate classes I've gained a significant piece of evidence to suggest that letting go and being creative in terms of teaching methodology can very productive.

I have a little S2 bottom set who are generally quite low ability.  I opened the course folder last weekend to see that I had to teach them negative numbers.  Hmmm - how can I teach these kids negative numbers in an interesting, visual and tactile way?

The answer was a two part lesson.  The first part was a powerpoint I found from somewhere which I could put on my smartboard.  It was the user interface of an ATM machine.  The bank account had £50 in it.  I had kids come up and "withdraw" certain amounts of money.  We then investigated what happened when people withdraw sums of greater than £50.  This led is onto the idea of over overdrafts and negative numbers in context.  The calculations of the overdrawn amounts seemed to come to the pupils very easily. Next we took a look at another powerpoint, this time we had a virtual thermometer and discussed temperature rise and fall.

So far, all very well.  But what about doing calculations proper, so to speak?  I always favour having a number line from -20 to 20 on the kids page and then they can do the "steps" method of counting along.  In a bid to break free of jotter work I printed out all of these numbers and lay them along our corridor.  I then got kids to answer quesitons such as -5 + 7 and - 2 - 4 by standing on a number and then moving the correct number of steps to get the answer.  We then had a girls v boys challenge quiz.  With the scores tied at 4-4 and first to five being the winner I opted to throw a spanner in the works, just to see how they would handle it.  I asked the girls 5 - (-4).  I got all sort of answers some of which were more contrived than others.  The girls didn't manage to get the answer, although at this point I didn't explain how to do the question.  The boys too didn't get the answer. However, upon me revealing the answer to them some interesting discussion took place.

Kid A: "Hey, does take away not mean difference?"
Me: "Yip". (Although I'm thinking to myself - where is he going with this?)
Kid B pipes in: "Difference, what's that?"
Kid C: "The distance"
At this point I gasp in astonishment as the kids point out that to get the answer of 5 - (-4)
all you have to do is count the number of "hops" between 5 and -4,  thus giving the required answer of 9.

Startling as it may be, this was a completely new concept to this particular mathematics graduate.  I suppose I'd always just accepted that two negatives next to each other become a plus.  The kids on the other hand had no preconception of this and as such were a blank canvas!  They made be really think that yeah, by old school definition take away is difference, so getting the answer 9 is perfectly logical!  I told my colleagues, some of whom are very experienced, and they had to admit this was something they too had never properly considered.

With some more questions answered, by using our corridor number line the kids managed to spot that the two negatives do indeed become a plus.

I have to admit this lesson exceeded my own ambitions as to how it might go.  It gave me a whole new insight into this particular area of maths and how to deliver it in future.  It's always been a dry area of the subject for me, with little scope for doing nice problem solving lessons leading to conclusions, like this lesson did.  However, in future, I will always be doing this lesson rather than just telling them the rule!

Some of my own preconceptions of the kids have been challenged and that is no bad thing.

In the spirit of the moment - I've planned lots of nice investigative lessons on the straight line next week for my S4 credit class.  I don't intend to stand at the board for very much at all other than explaining how to fill in a table of values for (x,y) coordinates given a formula. My hope is that they will discover and understand the gradient and y-intercept concepts by themselves based upon the task sheets I've designed for them do in their small groups.  Hopefully the outcome will be some nice posters by the groups detailing the key points of the straight line.  I figure that by doing it this way I may not need as much textbook practice, and maybe, we might get through the topic quicker than normal - with a better than average comprehension.

I'll let you know how I get on in my next post!

Embracing Chalk and Talk

The following is a loosely edited post I made on the Tes wesbite this morning.

I'm a young maths teacher - just starting my fourth year in the job. I agree that for some pupils lots of repetition isn't needed and for others a big page of examples is an effective way of helping pupils to commit a technique to short term memory. For recall of this knowledge lesson starters and homework's have their place. By recalling the facts fairly regularly the aim is obviously to commit them to medium term memory.

It is easy to think of practical activities for pupils to do when teaching perecentages: use a catoluge, get on the internet and look at ebay, amazon, finance websites etc. If I'm being honest I find teaching lower school far easier as the topics lend themselves much more freely to "interesting and practical" lessons. I'm not saying that it isn't possible to do interesting things with credit and higher material but it is more of a challenge. I am only teaching higher for the first time just now so at the moment it is the classic "here are the notes, here's how it connects to the previous lesson, these are some questions about it for you to try" all mixed in with some good questioning technique etc and lots and lots of homework. But it is very traditional and not revolutionary at all.

With this in mind I find it easier to comment on credit as I am teaching it for the second time through now. My fourth year set are doing algebraic fractions just now (they are the third out of four credit sets in ability terms). My question to fellow posters is am I wrong to just stick with the good exercises in the 4B textbook mixed in with some "challenging ones" I put on the board myself? I feel to deliver something like this I need to introduce the ideas at the board and "help the pupils to make the connections between exisiting knowledge and what we are doing now". They all know how to add fractions but do they know how to do it when denominators are (x+2) and (x +5) respectively? We could faff around and do this by means of a worksheet etc that leads them to the same conclusions as our class discussion at the board would do. My lesson just now would be a big discussion of various problems as a class and we'd try to solve them together. I woudln't just say "this is what you do". I'd ask them to apply existing knowledge while I write what individuals are contributing at the board. Once we have got this sorted as a class we then get it down into our notes for future reference. I think at the moment - they get to articulate what they are thinking - I try to vary who I get input from and they can inspire each other. If I were to try something more modern and fancy like a group trying a worksheet on the topic and with the worskeet leading them to the same conclusions; we'd get there eventually - only it would take longer and inevitably some pupils still wouldn't be able to get to the correct conclusion. After we've all understood a few examples at the board (by holding up between 1 and 5 fingers to indicate confidence levels) we'd move onto the text for the remainder of the period. By this means they can encounter lots of examples with a good gradient of difficulty and thus improve their own understanding and knowledge. Of course I could use loop cards or another sort of "primary or lower secondary" idea such as that - but I don't see why any other way of delivering the practice questions would be any better for the pupils than just using the text. I don't see an immediate flaw with this style of teaching for this class. It's very effective - my results have been continually good in final exams, prelims, homeworks etc. Also the kids do get to enjoy it - some of them are really relishing the challenge of the maths at the moment - plus we always get directed banter and class jokes when the teacher leads from the front - something that helps to create a nice atmosphere for pupils entering the room. What would other posters do differently? We have time constraints which mean we need to get to the correct conclusions fairly quickly so that we can practice them. I've thought a lot about content delivery at this early stage of my teaching career and know I've got lots to learn - but at the moment I don't see many more effective ways of doing it than that which I employ just now.

For a top set, it would be a very different scenario. When the kids are generally much brighter then it is more of a "sink or swim" scenario. I'd be more prepared to let them experiment and stuff. My top s2's manage to discover indicies lawys, pythagoras and all other sorts of stuff on their own - but I know the same lessons wouldn't have worked with less able pupils - I've tried before!

Basically - what I'm saying is that for more complex maths issues sometimes a traditional teaching methodology seems to be very effective with pupils who are good but not great. Only when you get a right good top set (and in our school we are fortunate that the top sets really are top notch) have I felt that the kids can "step on the shoulders of giants" by themselves, whereas other classes need a shove-up onto the shoulders from me! Any ideas, reactions, thoughts etc much appreciated. Feel free to criticise my naive attitudes - I share my own thoughts purely because I want to improve for the sake of the kids. Cheers.

First Post

I've often read blogs by other teahers and found that they invoke a variety of responses from myself including laughter, sage nodding of the head and contempt. I'm a regular reader of the Tes forums community.tes.co.uk, especially the Scotland-opinions area. There are often some good aritculate posts expressing lots of interesting ideas which are good to read. However, there is a lack of content specifically for scottish maths teachers. With this in mind I have googled a few times for "scottish maths teacher" to see if I could discover a blog by somebody with wonderful ideas on the teaching of my subject. I would love to read from people who have lots of persepective on the not just maths, but scottish secondary education as a whole and who have innovative and thought provoking ideas to consider. Sadly, however, I haven't found any such place.

With this in mind I decided that I, with my long and illustrious teaching career now entering it's fourth year should start a blog. I don't know what I'll gain from this. Maybe it'll be theraputic - but I think most of all I'd like to make connections with other teachers out there who do the same job as me on a daily basis and who really care about what they do.

Here goes. By the way, I forgot to mention my literacy is terrible, so apologies in advance.

Fiona Hyslop would not be happy.