Category Archives: #mixedattainmentmaths

When is a Quadratic “factorisable”?

There are 3 standard ways of solving quadratic equations once they are in the form:

ax² + bx + c = 0

They are:

  1. Factorise
  2. Complete the square
  3. Use the formula

I think I generally teach them in that order probably without much thought as to why. I guess the formula needs to be derived by using completing the square and factorising seems to follow on from multiplying out double brackets, which comes before all of this.  The I question that I sometimes get from students is “what’s the point in learning factorising if the two other methods always work?”.  Well, it’s quicker and you can do it on a non-calc exam is probably a standard response.

But have you tried using the formula without a calculator to solve a quadratic that you know will factorise?  Have a go.  Plug this:

x² - 3x - 28 = 0

into this:

download

…and solve without a calculator.

It’s a surprisingly satisfying experience, one that I would not want to deny my students.

You’ll need to know your square numbers because b² – 4ac will always give you a square number for quadratics that factorise. But the arithmetic is perfectly reasonable and is likely to be so for most quadratics that can be factorised.

When I did this recently, I had a great question from one of my students, as I was aching my brain trying to make up a quadratic that I knew would factorise.   “If you just picked one randomly, what are the chances that you would be able to factorise it?”

I’ve since had chance to investigate this further.  It’s a great question and there is a lesson in here, or at least an extension question to explore once the fundamentals of using the formula are secure.

I started approaching it by using the formula and focussing on b² – 4ac and what values of a, b and c would yield square numbers.  To simplify the problem, I started with a=1, so I was looking for when b² – 4c = 1,4,9,16,25, etc.

I then looked at it from the other end, i.e. starting with e.g. (x+1)(x+n) what values of a, b, and c are yielded.  Then vary further to look at (x+m)(x+n).  I just started with a few values of n and m to see if I could spot patterns.  I won’t spoil the fun by revealing those patterns, but this is very open-ended and could provide some intrigue for the right learners.

 

Fraction problems

These problems are ones that are made much clearer by drawing a rectangle to represent the “whole” and then deciding how to divide it into equal parts.  The numbers are not too tricky but interpreting the question might be:

Capture

These are not intended to be fraction of an amount questions.  An approach could be to decide upon an amount, but the intention is to direct students to drawing a representation of each question.

Capture

 

A perspective on teaching mixed attainment

In a world first for this blog, I have a guest post!

The power of collaboration and Twitter has led me to Bruce Gray (@bucksburnMaths)  who teaches in Bucksburn Academy in Aberdeen.  I am a recent convert to teaching mathematics in mixed attainment groups rather than sets but I am a pragmatist at heart and fully understand the reservations and difficulties with this approach.  It is not a small decision for a maths department to switch from teaching in sets to mixed attainment groups, especially if other subjects retain sets as my school does.  My summary of how to do it which I have written about before:

  1. Collaboration and co-planning must be part of it 
  2. Do it one year at a time, i.e. start in Year 7, and review each year

To find out more, check out www.mixedattainmentmaths.com.  The first #mixedattainmentmaths conference was held in January and plans are being put into place for another one in June.  If you are interested, leave a comment on this post or get in touch via Twitter or email.

Anyway, enough from me.  Grab a cuppa and read Mr Gray’s story.  Just, please – make sure you get to the end!

 

 

Mixed set maths classes. It’s a phrase in the past that has made my brain shriek in horror and caused me to physically shudder with trepidation. Throughout my time teaching (coming up to a full decade this year!!!) I’ve often wondered “Where would you even begin?” whenever the topic has been brought up in meetings and CPD events and pretty much every time I’ve came to the same conclusion: “Not a damned clue! I’m glad we don’t do it in our school!” My PT (Principal Teacher, i.e. Head of Department) had been talking about it for a while after seeing it at a Stirling Maths Conference (the highlight of any Scottish Maths teacher’s year!) but, much to my relief, nothing had actually came of it.

Then, at the start of the 2015/2016 session everything changed. I actually missed the first three weeks of the new school year. My wife (also a teacher) and I had our daughter on the first teaching day of the school year and I missed the same thing we did every year with our next S1 classes (equivalent to Year 7) – get them in, do some introductory lessons, slap a diagnostic assessment to see what they know, and then split them into the traditional “top”, “middle” and “bottom” sets before getting stuck into delivering content. I didn’t think anything would be different when I returned as “That’s the way we’ve always done it.” Little did I know that I was going to go back and everything had changed! Much to my dismay I returned from my three weeks of paternity leave to discover the department had decided to bite the bullet and take the plunge into the world of mixed set teaching.

I can’t remember exactly what my reaction was to this news but it was not wholly positive – I’m imagining a combination of an ashen faced “You did what?” and a sinking feeling in the pit of my stomach. The way the S1 classes had been timetabled to come to Maths meant it wasn’t really possible to set them all in an effective way and would have resulted in a couple mixed ability classes so, in order to give all pupils the same experience, we were just going to keep them all in their register classes rather than in ability groups. A sound reason. The fact my colleagues had discussed it at length and believed that it result in a better standard of learning experience for all pupils also reassured me so I decided to stop moaning, roll my sleeves up, and get on with it.

I hated it.

Serious, genuine, unbridled hatred for the whole damned thing.

I tried my best to be positive, I tried my best to make it work, I tried my best to deliver lessons that meet the high standards I have always set for myself and it felt like every single time I tried during those first few months I failed. Completely, utterly and hopelessly. I remember my misery coming to a head one rainy Winter morning. I was running over my latest set of failed lessons in my head during a 6 a.m. session of making baby’s bottles before going to work when something inside snapped. “I don’t want to go to work. I hate my job” I muttered to myself through a choked back sob with slumped shoulders and eyes reddening with tears. The whole thing had made me totally and utterly miserable. I had no idea what to do. For the first time in my career I had serious doubts about my future. I’d always said that if the job ever became a chore, if I ever fell out of love with it, then I would throw in the towel and find something else. Rather than doing anything silly – like quit my job there and then or hide in the stationary cupboard until the whole thing went away – I had a talk with my PT about it. I am really lucky and have always appreciated his approachableness and the support he’s offered over the years. I was glad I was able to go through the cathartic process of getting it all off my chest and it helped me break the problem down and see what the actual issues were. As far as I can remember there were three points that made those mixed classes the bane of my existence.

 

1) Organisation

Organisation has never come easy to me. It’s something that I’ve had to work hard at. In recent years I think I’ve been a bit guilty of getting complacent in terms of my organisation and fallen into a routine. I’d gotten to know what each level of class should be working on, how I’d go about delivering the content and what resources I’d use as part of the learning and teaching. Building this bank of knowledge up has really helped me keep on top of my day-to-day and long term planning but it also meant that when faced with something new (like this new system of mixed ability classes) I was lost. The rug was well and truly pulled out from underneath me. My safety blanket of experience was gone and it meant most lessons had to be planned from scratch – something I really wasn’t organised enough to do.

 

2) Workload

I direct fallout from the above point. Having to plan a new S1 course as it was being delivered required a lot of work all the time. I felt that as soon as I had something planned that I was happy with I had to get on with the next topic – there was no time to stop, collect my thoughts and reflect on what worked as we had to hammer on with the next set of lessons. The big thing that is needed to make mixed ability lessons work is differentiation and while this often takes place in my non mixed set lessons it needs to be done in a much more meaningful and effective way. A greater range of activities needs to be planned which leads to a much heavier workload. Matters were not helped by the fact that I was usually reluctant to use resources I’d found online and instead preferred to use resources of my own. This really was unsustainable.

 

3) “Won’t somebody think of the children!?!”

I may make light of it with that heading but this was a real sticking point for me. At every meeting we had as a department to discuss how we and the classes were getting on with this new set up I would constantly be bringing up the fact that I was hating it. Everything I said would usually start along the lines of “Look, I want this mixed ability set up to work BUT…” I meant the first bit, I really did want it to work, but I knew I was not meeting all the needs of every child in my class. This was something that really hurt to admit to. I didn’t feel I was doing a good enough job with the children at the extreme ends of the ability scale by aiming “right down the middle” with my teaching. The higher attaining pupils weren’t being stretched at all and the less able pupils found the work too challenging. This was the aspect that made me feel the worst – the fact that I perceived myself to be failing in terms of delivering an education and experience that each and every child has the right to. If you’re not doing that then you quite frankly aren’t doing your job.

 

Unloading and actually putting into words what was making me miserable was a good starting point to actually addressing the issue however it wasn’t something that was really possible to fix on my own. The department I work in is one I am passionately proud of and a huge part of that pride is down to the people I work with. I may have more in terms of “years served” in the profession than most of my colleagues but whenever I watch any of them teach I almost always come away going “I wish I could do that as well as they do.” and I have nothing but the utmost of respect for them. They are the reason I didn’t jack it in and quit when I had my crisis of faith. There is no way I’d willingly leave a team like the one we have. Things didn’t suddenly get better overnight and it took time me to be able to even say “mixed ability classes” without wanting to spit but I personally, did get there. The three things that helped me get there:

 

1) Divide and conquer

This is key in combating the workload. We had five mixed first year classes all following the same course and everyone planning different things for them. It was a colossal waste of teacher time with all the development work being needlessly repeated. The way around this grew quite organically. I found myself talking to my colleagues more and asking things like “What are you planning?” and “How are you going to deliver this part of the course?” Then I found I was given a well needed boost in confidence by being able to say “Oh, I’ve got something like that already.” and firing off an email with the activities attached. It actually got to be quite exciting opening emails and finding a bundle of emails with new activities ready to use. What ultimately helped me get over my reluctance of using other people’s resources is the fact that I have a deep set respect for my department as both teachers, work colleagues and friends and I trust them wholeheartedly with each and every aspect of the job. Each and every one of them passes the test of “Would you want them to teach your child?” with flying colours and it’s an absolute privilege to work with them. This job really is much, much easier if you are surrounded by good people! Our planning system developed beyond the scattershot of emails being sent back-and-forth and got more organised with everyone taking a topic, collating resources and developing a differentiated plan for the department to deliver. I’ve really grown to enjoy this. The fact that the resources for the current block of work has been prepared in advance frees up time for you to really focus on the next block and really look at what you want to teach and how you are going to differentiate the resources. In terms of what the pupils are doing it really gives consistency across the department. It means all the pupils are covering the same content and getting the same experience.

 

2) Everyone needs to buy in

This is vital for mixed ability classes to work. Every member of the department needs to be willing to do their share of the prep and to give the teaching of mixed classes a go. I found it much easier to deal with and was willing to give it a go (despite the fact I absolutely loathed the system) because everyone else was in the same boat and me refusing to even try to work as part of the team would just let everyone else down. I find the pressure of “Oh no! Everyone is going to see what I’ve prepared!” a bit of positive pressure. It means I have upped my game and made sure I’ve put a variety of different activities in my blocks of work. It also means I’m doing things that I would never have done with my classes before. I would have balked at the idea of something like a “treasure hunt“ or something similar but, after being taken out of my comfort zone with these classes, I now I find myself willing to give these a go with my set classes.

 

3) Be willing to try new things

You can’t be precious about the way things have been done in past. This is something I’ve been guilty of before. I’ve often uttered the phrase “Well, I’ve never done it like that before.” when faced with an alternative way of teaching something before trotting out exactly the same lesson as I’d delivered the same time the previous year. I feel (actually that’s not strong enough – I know for a fact) teaching mixed ability has made me a better teacher. I’ve had to think more carefully about what I teach, how I teach and the kind of activities. I also feel I had become a bit complacent in my teaching. I didn’t feel like I was an expert but I definitely was a bit arrogant in terms of “My way is pretty damned good and there isn’t really much point in changing things.” I’ve now realised that it can be a very freeing experience to burn everything down and start building from scratch (ought it’s not something I’d advocate doing too often!). Another thing that has helped is relinquish control (to a degree). Letting others have control of your resources is part of this but giving pupils more freedom over the tasks they are doing has been a very enlightening experience. For mixed ability teaching to work there needs to be a range of differentiated materials provided for most lessons. I assumed there would be a large group of pupils who would always go for the easy route and would pick the straightforward tasks every time. The truth of the matter is I am continually genuinely surprised at the kinds of comments my pupils make. “I’m going to try the easier sheet but if it’s too easy I’ll come back and get this tricky one”, “I think it’s going to be tough going but I’m going to give this challenging option a go” and similar such comments are regular occurrences and while there is the odd pupil who will go for the easier activity rather than pushing themselves I am often surprised by the maturity and insight the kids show when given this responsibility for their own learning.

 

While I feel I am a convert to the church of mixed ability teaching I would not say I’m a believer who follows its teachings without questioning. Maybe it’s just my suspicious nature but I view each new topic warily and mutter “How this going to work then?” to myself before diving in. I’m only half-way through my second year of teaching this way and I can still see potential dark clouds looming. Whether or not they actually come to anything remains to be seen but it would be foolish to ignore any of these potential problems. Again, I could break these issues down into a trinity.

 

1) Resources and budget

It turns my stomach whenever I hear of cuts to education be at a national, local authority or individual school level however, despite the fact that education should be viewed as an investment rather than a black hole money is just poured in to, these cuts are a very real and current issue which means we are expected to do exactly the same job with less funds to go towards day-to-day items such as basic stationary, photocopying and printing budget and the purchase of resources. A major factor that has made our mixed ability teaching achievable is that we have moved away from textbooks (for the most part, there are still times where a good, well written textbook is the best resource for the job in hand) to differentiated worksheets, card matching activities and active learning tasks all of which rely heavily of having a healthy photocopying budget. We are trying to make savings by reusing resources but that is not always possible and it is inevitable that new copies of resources will have to be made. The fact our class sizes are small has also made progress with this system much more manageable. It goes without saying it’s far easier to get around a class of 20 individuals than a class of 30+. I know we’d cope with bigger class sizes but I don’t think he pupils would get as rich an experience and from a classroom management side it would be that bit trickier to administer. Our teacher numbers and class sizes look safe for the future but it is still a very realistic concern to have.

 

2) The extreme ends of the ability

I think we are doing a pretty good job of catering for a majority of the pupils in the classes as the lessons are well differentiated or the content is delivered in such a way to make it accessible. There is still a nagging doubt though that some pupils don’t have all their needs being met. It’s easy to worry about the low end pupils (and when I think about differentiation my mind automatically goes to “Right, I must plan something for the less able pupils”) and it’s easy to forget the high end pupils or to just give them more work rather than something truly challenging. I don’t think that this is a problem inherent to mixed ability class. Even with classes set by ability you will have individuals that fall out with the main group. Now I sit down and organise and type out my thoughts it seems less of a problem and more something to keep in mind and not let slip.

 

3) What effect will it have long term

In the short term the mixed ability classes have been a hugely positive experience. There is a genuine buzz in the department when we have these classes in and I think it’s making me get better at differentiating with classes across the board. It’s got rid of the classic problem of kids being turned off from Maths because “I’m stupid – I’m in a bottom set, I’ve always been in a bottom se and always will be so there’s no point in trying to better myself.” Sure, there’s kids that still don’t like the subject – you can’t please all of the people all of the time – but comments of “I can’t do it.” are gradually being replaced “I’ll give this a go” and “This looks tricky. I’m not ready for it… YET!” It’s really encouraging to hear little comments like that being made. It’s part of the reason I get up in the morning! The long term effects remain to be seen. My gut feeling is that it will help attainment. If pupils are engaged and motivated it feels like that has to be the inevitable outcome but we can’t say that for certain without cold hard facts so we’ll just have to wait for the dreaded benchmark of “What did they get in their exams” a couple years down the line before labelling our journey into mixed ability teaching a success.

 

In conclusion… Well, I’m not sure I have one. Do I still hate mixed ability teaching? Is there still the “Serious, genuine, unbridled hatred for the whole damned thing” I mentioned at the start of this ramble?

No, not at all!

Hate has given way to something else. Love? No, not that but thoughts along the lines “I’m can’t do it” have definitely been replaced with a gleeful “I’ll give this a go.”

 

Pattern sniffing with Decimal Subtraction

An idea for a mixed attainment class that came to me about 5 minutes before a lesson today:

  1. 3.4 – 3.04
  2. 5.2 – 5.02
  3. 7.8 – 7.08
  4. 8.2 – 8.02

Find other questions like this.  (The “weakest” student in the class told me the pattern before I’d even finished writing the fourth question on the board.)

What do you notice?  Why is the answer to Q2 the same as the answer to Q4?

Can you create a question with 0.54 as an answer?  How many different answers are there to these types of questions?

Then:

  1. 5.7 – 5.007
  2. 8.3 – 8.003
  3. 6.4 – 6.004
  4. etc.

These are more tricky and test the skills of column subtraction, something that should be secure by Year 7 but may not be. Maybe an opportunity for collaboration amongst students to show how.

And then finally, try these two calculations. Which is easier and why?

7 – 1.392

6.999 – 1.391

Show on a number line why this works and then try some more.  I think these questions are interesting to explore.  But I would hesitate to recommend it as a must-do method to solve e.g 8-2.5687.  Whether or not it is easier to turn it into 7.999-2.5686 or not is an interesting discussion and one which I would want my students to form their own opinion on, not be too swayed by mine.

 

 

 

Collaboration

collaboration-mindset
I’ve had a couple of experiences of collaborative lesson planning at different schools. What we are doing at the moment with our mixed attainment Year 7 classes feels like the best I have experienced. I’ve been pondering why this is.

Collaborative planning can be a hard sell .  It can feel inefficient.  Some may be thinking “I could have planned this lesson in the time we’ve been sitting here discussing it”.  It’s sometimes difficult to get everyone involved, especially if the less experienced teachers perceive that the more experienced teacher will know the “best” way to teach a topic and therefore don’t feel they have much to offer.  It’s taken me time to appreciate that there is no “best” way to teach anything – it’s a complex interplay of the teacher’s experience and preferences, the relationship with the class and what’s gone before in the sequence of learning.

However, it is also patently inefficient for 6 teachers teaching the same topic to the same age children to be creating 6 sets of powerpoint slides and worksheets.  This struck me when I started teaching. How much time do teachers spend creating their own resources and lessons as if they were the first person to have ever taught it!?  At the time I thought why doesn’t just one person in the department to it, or, better still, one person in the country.

If classes are in sets then there is a justification that each teacher needs to “tweak” things for that particular class and the “level” they are working at. I would speculate that this is usually down to the teacher’s preconception of what the class will or will not make a decent stab at rather than a clear understanding of the individuals’ prior attainment with that topic.  However even if tweaking is required (we still tweak for our mixed attainment classes) it is still a whole lot quicker than starting from scratch.

There are, of course, plenty of lessons resources available either for free or by buying into schemes such as Boardworks, which could mean that nobody has to create any PowerPoint slides ever again. Or your department could follow a text book scheme which often have on-line resources to support them. But picking up someone else’s slides and just stepping through them is unlikely to result in a good lesson, teaching is not as simple as that.  It’s not really about the PowerPoint.  There are many subtle nuances within the flow of a lesson that are nothing to do with what is on the screen, the worksheet or the text book. And the temptation to spend to glance briefly at the resources just before the lesson without really thinking properly about the sequence of learning is always there.

This year, our Year 7 teachers have been meeting once a week for 55 mins in a timetabled PPA slot we call a TRG (Teacher Research Group).  We are doing the same for Year 8 teaching. It is voluntary for teachers to attend these TRG sessions as it is not additional PPA time.  So far, all teachers do regularly join and generally feel that they get back the time they give up.  There is one designated lead teacher for each year group who “chairs” the TRGs and is responsible for keeping the files and folders organised.  Being the lead shouldn’t be a big burden but it is important to have someone clearly identified in this role.

The TRGs vary, but generally will start with looking at a very simple overview of the next 8 lessons or so with names and dates alongside which we might spend a few minutes discussing – effectively the medium term plan.  We try to keep a week ahead of the lessons being taught.  Too far in advance and we are likely to forget what was discussed.  Going the other way, we risk the situation (which we have sometimes slipped into) where lessons are shared the night before and we miss the opportunity to discuss them.

It is this discussion of the individual lessons that is the crucial part of co-planning. The teacher that has planned the lesson will run through it, with the other teachers thinking about their own class and picturing how things will run. As we go, someone makes changes on the slides or writes brief notes on them to follow up later.  The discussion focuses on specific explanations and questions, but can also cover the flow of the lesson, the practicalities of classroom management or verbal questioning.  Often there is not a clearly defined end point for a lesson, it is up to teachers to manage the pace with the class and draw the lesson to a close at an appropriate point. This can mean that we sometimes get out of sync by a lesson or two but this is manageable.  We go at a decent pace in the TRG which enables us to get through a week’s worth (4 hours) of lessons.  We have 6 classes in each year group, so that means that each teacher is creating a lesson just under once a week.  The starting point for this teacher is sometimes last year’s lesson, or sometimes we will briefly discuss the key learning points and key questions for the lesson in the TRG to give the teacher planning it enough to go away with to start planning.  Homeworks are also created collaboratively and are linked closely to the lessons aiming to include some more challenging open-ended or problem-solving questions alongside practice of core concepts.

In addition, we do “learning walks” once per week where one class will be covered, usually by the Head of Department.  The teacher will go round the other classes to see the lessons in action.  They are looking to see how the pupils are responding to lessons we have planned and are not commenting on individual teachers. Every few weeks we will devote some of the TRG time to learning walk reflections where we discuss more generally what we have noticed and what we need to change.

Personally I have learned a lot from our TRGs this year.  As well as an efficient way to plan high quality lessons, it is also a very effective form of ongoing CPD. I feel more confident actually teaching the lessons having had the chance to discuss them with colleagues beforehand. It has been a fundamental part of our transition to teaching in mixed attainment groups, which I don’t think would have worked without collaboration.  I also notice that we have more informal reflective discussions between teachers about how the lesson went as we all have the common ground of having taught the same lesson.

The key challenge of course, is time. It would be great to continue this model right up to Year 13 but timetabling all these TRG sessions will be difficult. My message to anyone in a position to make a decision on this would be to have a go, make a start.  If TRGs are effective then they deliver high quality lesson planning and ongoing CPD for all staff involved.  These are two pretty fundamental aspects of teaching, aren’t they?  So let’s dedicate the time to them.   Marking policies might be a place to look at for clawing back some of that time.  But that’s a topic for another post…

 

 

Bus Stop Division

Here’s a big number:

screen-shot-2017-01-31-at-06-48-48

Try different single-digit divisors.  No remainders.

This is an example of purposeful practice – exposing the wonder of mathematics whilst providing a reason to practise lots of  of bus stop division.

You might want to start by asking pupils to come up with their own dividend “in the tens of millions” and try different divisors. (Here for a quick primer on the mathematical language.) Inevitably they will end up with remainders, which they may or may not carry into decimal places. Then let show them this “magic” number.

Questions to ask:

  • What divisors does this work for and why? (Purposeful practice)
  • What other dividends could I make like this? (Purposeful practice + reasoning)
  • What smaller dividends could I make like this? (reasoning)
  • What is the smallest dividend I could make that all numbers 1-9 will divide into without remainders? (reasoning)

Whilst I would want everyone in the class to understand the reasoning through a whole-class discussion, you may have some learners who need the practice on bus stop long division and spend most of their time doing this. Those that are confident with this technique can spend their time exploring deeper into the structure of the number.

Whilst we are on the subject of “Bus Stop”, maybe this technique actually has nothing to do with standing in line waiting for a bus: