Author: tombennison

Earlier this evening I saw the following picture posted by Mathster (@mathstermaths)  

 
As I tweeted I am really not keen on the use of “move all terms to the left” as I don’t think it is clear what this is meant to mean mathematically. Indeed, it could easily lead to misconceptions with people thinking that you just move terms over an equals sign with no conceptual understanding of what should be happening. Stephen Cavadino (@srcav) has written about this here. 

I thought I would try going online and doing the same equation as is solved in the example. The step by step solution I obtained was even stranger: 

 
I have no idea why there are steps 1 and 2 simplifying the right hand and left hand side before the pointless step of rearranging to obtain everything equal to zero…….

Christine (@MissNorledge) also pointed out the strange behaviour this app shows when rounding. According to the example above 8/3 rounds to 2.666666666666665. To me this indicates poor use of floating poor arithmetic. 

Jasmina then showed something that is clearly a bug in their parsing library  

 I’m curious as to how this app is working, specifically

• How the user input is being parsed?
• Are they using a JavaScript symbolic algebra library (and if so which one)?
• How is the “maths” implemented?

Unfortunately at a first glance the JavaScript viewable using the Safari developer options isn’t particularly illuminating. As I’m away for the weekend I haven’t been my able to delve very deeply into the code with a good editor like Sublime and work out what is doing what. 

The app can’t solve quadratics with complex roots or cubics: 

  

 All in all I want to know more about this…

I’m finally getting the material from my session online – if you want an overview of #mathsconf5 last sturdy check out my post here.

In this post I am aiming to share the materials I handed out in my session “How to Enjoy (and Succeed at) Your NQT Year”

I’ve embedded my Prezi below, if you were at my session you would know that I didn’t get anywhere near through it.

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All the materials can be accessed on my website at the links below:

• Handout
• Starter Task Sheet
• Mathematics Questions
• Colin Foster Article

I started the session by finding out the strengths and weaknesses of attendees. In my opinion you will have a good understanding of these by the time you start your NQT year (even if you are an ITT student I think you often know what they will be before you have spent much time in the classroom). For me, my strength is that I am pretty good at stretching the high attainers whereas I see behaviour management as a weakness as this is something that I have had to work quite hard at to get where I am now. Being frank with yourself about your personal qualities is very important if you want to improve and develop your practice.

Following this we talked about behaviour management strategies; I was very interested in some of the conversations happening about the use of reward lessons in the computer room for good behaviour amongst other things. My top tips for behaviour management is to ensure that you are always consistent (however tired and fed up you are) and follow through with any sanctions you hand out. It is very easy to let things slide as term goes on – especially in the winter months when it is dark and cold – try your best to not let that happen.

Moving on to lesson planning I talked about my five key things to consider when planning a lesson:

1. The big picture.
2. The key learning objectives.
3. 5 key questions.
4. Any possible misconceptions.
5. How will you assess learning.

For me the questioning used with the class and individual learners is the most important thing to bring on the mathematics of students by prompting them to think critically about it. This fits in nicely with a school focus on questioning in my school which is good. Our new Year 7 scheme has questions for most lesson plans. I think considering questions before the lesson means that I think more in depth about the maths that I am presenting and how it will come across to learners. The same can be said for considering misconceptions; some of the best questions in my opinion will draw out misconceptions and enable us to tackle them head on as a class. I was possibly a little controversial in this section by saying that I don’t really like the assessment being on lesson plan pro-formers. This is not because I don’t think it is important – it is of course crucial to know where your students are with the topic of a lesson! My point here was that I don’t want how I am going to assess my students (which is really my problem to resolve) to affect how they experience the mathematics that we are doing that lesson and so my mode of assessment shouldn’t be a focus of the lesson planning process.

Many teachers now have to move rooms between lessons. When this is the case I find that it helps to have some go to starters that don’t require me to have logged on to a computer and turned a projector on. For this reason, this year I have stuck a sheet with some easy “no-plan” starters into the back cover of all of my lower school student’s books. I have written a bit more about this in my post “No Plan Starters”. With practice I find that the students don’t need too much guidance if I just tell them a starter task (and maybe write a few numbers on the board) and it means that they are practising some basic skills while I am waiting for the computer to load.

Before I was planning on discussing the advantages/disadvantages of technology in lessons I talked about homework. My guiding principle is the I want a homework that takes the minimum amount of time to prepare and mark but that gives the maximum possible benefit to the students. A type of homework that fits this bill are the PRET homeworks as shown below.

I particularly like the research and stretch components of this style of homework. I found that last year, being very precise with the expectations for the first couple that I handed out meant that students soon learnt what was expected when I gave them a homework in this style. Before you go away and write some you should definitely check out Jo’s (@mathsjem)and Kathryn’s (@DIRT_expert) PRET homework website as there are plenty already to use!

For technology in the classroom I was going to talk about different tech tools so I think I shall write about this topic more over the next few months….

I had a large section planned entitled “Do Some Maths”. This is actually my number one tip for your NQT year! It is all too easy to get caught up in the mundane tasks of a teacher that you don’t spend time doing any maths apart from the questions you need for lessons. I’m sure that most of us chose to teach maths because we love doing maths so I think it is important to spend some time doing maths that is challenging. Here is a nice question:

A particularly nice source of challenging problems is the book “Calculus for the Ambitious” by T.W. Kernel. This is a book targeted at strong A-Level students giving a more university-like approach to Calculus than you see at A-Level.

Your NQT year is an ideal time to develop your subject specific pedagogic knowledge. Use those extra frees to go an observe other teachers focussing in particular on how they teach a given topic. Look for how they structure a lesson to build understanding gradually, how they tackle topics that are perceived as being hard to teach and how the activities they choose promote confidence in the learners. Another excellent way to build your subject knowledge (both pedagogic and general mathematical) is to join at least one, if not more, subject associations. Personally I am a member of the ATM and the IMA and value the access this gives me to their journals. On the education journal note it is definitely worth getting involved in the #mathsjournalclub discussions that I organise – the next one is on the 19th October, see here for more information.

On the subject of marking, which is often seen as a necessary chore, and any time you spend doing it should have an impact – it definitely should not take over your life. Kev Lister’s (@listerkev) RAG123 marking has revolutionised the way I mark. Being able to get quicker feedback from the students is really helpful, it doesn’t take that long to get through a set of books commenting if necessary. I do a deep mark in line with my schools marking policy in addition to this, but having looked at the books so much these deep marking sessions are much quicker than they would have been.

I believe that having a network of teachers you can turn to for advice outside of your own department and school is incredibly helpful. Twitter makes this easy, so spend some time on twitter and take part in as many twitter chats as you can. I provide a summary of all the maths specific chats here.

My plan was to end on giving some tips for your NQT year from people other than me:

• Don’t do any work on a Saturday.
• Follow up on sanctions – but make sure they are fair.
• Always stick to class routines that you set.
• Don’t be afraid to say when you are struggling – someone will help you.
• Laminate resources.
• After a bad lesson find a positive – even if it is that you kept all the pupils in the room.
• Always remember the good pupils and how much you enjoy teaching them.
• Write any advice given to you down.
• Divide your time away from school into working time and non-working time and stick to it.
• Talk to anyone and everyone.
• Don’t overdo it

and then mention a few things to avoid including listening to people moaning all the time and working all the time.

The main thing I have learnt myself from this session is to not plan as much!! Thanks to Danny Brown (@dannytybrown) for his interesting points in the session and to Rob Beckett (@RBeckett_Yd) for having a quick look over the slides previously and giving me an NQT’s perspective on them.

Tonight we had the first of the East Midland’s branch of the IMA evening talks. David Worsley, from Network Rail (and a Visiting Lecturer at Newcastle University) was talking about mathematics in the rail industry.

It was a very engaging, entertaining and informative whistle-stop tour of the various mathematics used in the rail industry.

He started by talking about the forces on rail bridges before moving on to talk about the cant of a track. I had never considered the fact that if rail track bends were not banked then the outside track would experience greater wear than the inside track resulting in a need for an earlier replacement. This, as well as the desire for passengers to experience little centrifugal force explains why a cant is put in on the bends. The cant would be perfect if, in the picture below, \(a l_j\) was zero. However each piece of track has many different trains running over it at different speeds so selecting the optimal cant is an interesting optimisation problem. 

 Seeing a table showing how the Department for Transport values the time of people commuting by various means  

 If you are a passenger in a taxi your time is worth more than anyone else. 
The discussion concerning strategic optimisation and  the planning of the HS2 route through the midlands was very interesting as it is an issue that is local to us. 

 If you fancy checking out the talk there is a video here. 

Thought I should provide an update on last nights post.

This morning the MyMaths team must have come into a deluge of tweets as they started work tweeted that they apologised for the message on these slides and were working to update this lesson.

True to their word, by the afternoon this lesson had been updated and the first slide is now the following

They have also published this blog post about when they update content. 

While I don’t think it was very good that the original slide ever existed I am very impressed with how quickly they have acted on our feedback and changed their slides.

Thank you for doing this – it is good to see such a negative message removed.

I’ve written about a pretty awful MyMaths question before and this is another post about MyMaths – I promise I will keep it brief.

When I saw the screen shot below that had been tweeted by Michelle MacDonagh (@michellemacd) I genuinely thought it was a photoshop by someone. Alas, it turns out that this is not the case: 

For a product aimed at the maths education market to promote this idea in any way is unbelievable.

To be fair to MyMaths, clicking through reveals the next part

Personally, I don’t think this mitigates what is at the top of the page. Many students will be looking at this lesson on their own, and perhaps not pick up on the obvious untruths that have been written above.

Unfortunately, clicking next one more time reveals this ridiculous pearl of wisdom from MyMaths

What a great way to promote the importance of understanding behind a concept…

I cannot think of any justification for this… I would be very interested to hear MyMath’s reasoning behind this.

Yesterday morning I wasn’t as positive setting off early to #mathsconf5 in Sheffield as I normally am due to having felt really ill on Friday and I still wasn’t feeling great on the Saturday morning. Giving a workshop in the first session meant that not going wasn’t really an option and I was very thankful to @Miss_J_Carter for offering to drive. I’m also going to put not being sure if I recognised Hannah (@MissRadders) in the queue for car park money down to my poorly-ness!

When I arrived I was (incredibly) excited to be given this tea towel by Julia (@tessmaths):

I hope she understands just how happy this tea towel made me!

The conference started as usual with a presentation by Andrew Taylor of @AQAMaths – he helpfully put his slides online here. I was particularly interested to hear about the KS3 assessment materials that they are going to be releasing.

Following that we had the speed dating 🙂 The first one ended up being more of a general chat though. For the second someone who isn’t on Twitter shared the idea of having a grid that students work through to formalise the process of solving an equation for example. I like this idea as it provides a bit of structure and would hopefully lead to better written mathematics from students. The last person I spoke to is also not on Twitter and talked about using tools such as Secretive which I have used a couple of times before but then also mentioned Mentimeter which I haven’t used before. A friend of mine recently mentioned the graphing capabilities of Mentimeter so i need to look at this soon.

I’m going to briefly highlight the best bits of each session, but I didn’t take as good notes as normal so they are definitely not comprehensive!

Session 1: 

I was delivering a workshop entitled “How to enjoy (and Succeed at) Your NQT Year” during the first session of the day. I’m not going to say a lot here as I plan to write about it specifically (hopefully tomorrow, definitely by Friday) but I enjoyed the discussions I had with people attending my workshop. It was particularly nice to have @dannytybrown in my session.

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Session 2:

For Session 2 I had chosen @dannytybrown‘s workshop “Time to Slow Down?”. It wasn’t what i was expecting but I really enjoyed it. I was sat on a great table where we had a lively discussion about the initial slide 

What do you notice? I had never really considered looking at the formula for the sum of an arithmetic sequence in conjunction with a picture of a trapezium before – some great points were made by people in the audience.

We then discussed an account of a conversation between Danny and someone else about an observation of a particular A Level lesson. The conversation is hopefully viewable in these photos of the handout:

  

It was interesting to consider this as it raises many important questions. I certainly think becoming emotional and letting yourself be annoyed means that as a teacher you don’t reflect properly on a lesson.

Session 3:

Luke Graham (@BetterMaths) chaired a session where attendees collaborated to come up with a list of topics that they are wanting ideas of how to teach in the new GCSE and then some ideas of how to do this. I had been asked to oversee a table looking at the Statistics strand – there really isn’t much in the new GCSE on statistics. The result of this session is shown in Luke’s tweet

I’m looking forward to seeing where he takes this.

The TweetUp:

During a nice long lunch break ( the mini pies were pretty good) the TweetUp organised by Julia (@tessmaths) took place. This is becoming a bit of an institution and I was on the QRCubed table again which embarrassingly I had forgotten how to solve. Lots of people were getting involved with Pete Mattock’s (@MrMattock) STEP questions, Jo’s (@mathsjem) selfie booth and the various other activities on offer. The “TweetUp and do Some Maths” is a great part of the day and if you haven’t been to one before make sure you come at #mathsconf6.

Session 4:

This was the first session that I decided on. Bruno Reddy (@MrReddyMaths), Craig Jeavons (@Craigos87) and Matt Fox (@MFx15) have recently returned from Shanghai and this session “Sh****ai Is Not ADirty Word” was dedicated to them sharing some of the things they experienced and noticed.

I was interested to learn that the students do eye relaxation techniques every day – perhaps something we should implement over here, I’m sure it would help me at least! I think the fact that teachers in Shanghai work in 5 year cycles and must log 240 hours of CPD in those 5 years in order to progress to the next level is significant. This is much more than in the UK and surely improving teacher’s practice is the thing that will lead to the biggest increase in assessment scores (whether we want this to be the focus is another question!).

The quick ramp up in difficulty of the questions concerning the simplification of surds shown on the slide below is very interesting. Question 3 is significantly harder than question 1.  
I would be afraid to use such varied questions (where the difficulty increases quickly) in a “quick response” activity as shown below.
 I liked the evidence of students who are equivalent to Year 10 using the \(\because\) and \(\therefore\) symbols to explain their mathematics. I firmly believe that we should be encouraging this sort of formalism in our students, certainly at A-Level.

  

The variation in the index law questions above is striking. In the UK we would normally do lots of similar questions such as:

 Because of this our students aren’t being exposed to as wide a variety of situations where the index laws can be applied – this leads to difficulties in the future, for example when cancelling algebraic fractions down.  James Pearce (@MathsPadJames) has written about this in a great post yesterday.

Session 5:

Kris Boulton’s (@Kris_Boulton) talk on the “Stories of Maths” had been one that I was really looking forward to and he didn’t disappoint. Writing about it really can’t do it justice and luckily I recorded it, so if you missed it download the file (it’s about 1.6GB) from here and watch it – you won’t be disappointed.

I particularly enjoyed the reading about Sumerian (I think!) shepherds and how they could find out if all sheep had returned before the invention of counting. Will definitely be using that in class. The origin of the word “sine” was also interesting as I didn’t know the full story there.

Overall it was a really good day, and as ever great to catch up with people. Mark McCourt has written a blog post outlining the philosophy of the mathsconfs that is worth a read.

Looking forward to the next one on the 5th March in Peterborough…..

I was pretty excited to discover that my local Aldi had Romanesco cauliflower’s for sale – from Nottinghamshire no less.  

    
 
Above are some pictures at various levels of zoom. 

I really like looking at Romanesco cauliflower as it is possibly the closest approximation in nature to a mathematical fractal. Because of nature’s many approximations to fractals, fractal growth is incredibly important in computer games to create life-like environments. 

If you are interested in coding a simple fractal check out my old post on Barnsley’s Fern. 

So the day to announce the next #mathsjournalclub article as come, and it was a landslide victory, garnering over 52% of the votes.

The article you have chosen is “Mathematical études: embedding opportunities for developing procedural fluency within rich mathematical contexts” by Colin Foster, as published in the International Journal of Mathematical Education in Science and Technology.

The abstract is reproduced below and the article can be downloaded by clicking on this link.

• In a high-stakes assessment culture, it is clearly important that learners of mathematics develop the necessary fluency and confidence to perform well on the specific, narrowly defined techniques that will be tested. However, an overemphasis on the training of piecemeal mathematical skills at the expense of more independent engagement with richer, multifaceted tasks risks devaluing the subject and failing to give learners an authentic and enjoyable experience of being a mathematician. Thus, there is a pressing need for mathematical tasks which embed the practice of essential techniques within a richer, exploratory and investigative context. Such tasks can be justified to school management or to more traditional mathematics teachers as vital practice of important skills; at the same time, they give scope to progressive teachers who wish to work in more exploratory ways. This paper draws on the notion of a musical e ́tude to develop a powerful and versatile approach in which these apparently contradictory aspects of teaching mathematics can be harmoniously combined. I illustrate the tactic in three central areas of the high-school mathematics curriculum: plotting Cartesian coordinates, solving linear equations and performing enlargements. In each case, extensive practice of important procedures takes place alongside more thoughtful and mathematically creative activity.

This looks a really interesting article and I hope that many of you will join us (despite it being term time) for the discussion on Monday 19th October at 8pm.

The second and third place articles will now go through to the next poll, along side some other suggestions.

I hope you enjoy this article!

One of the many jobs that I had wanted to do over the summer was to create PRET homeworks for the A-Level modules that I was going to be teaching this coming year. Along with the website design this didn’t really happen, so now I am doing them throughout the year as I go.

I thought I would share one I made this week as I am trying to do at least one resource post and one more general math/education/math education post every week.

Jo Morgan (@mathsjem does a great job collating all the PRET homework that people contribute at her PRET homework site, so make sure you take a look to see all the wonderful homework for KS3,KS4 and KS5 that are available. The one I contributed this week is for “The method of differences” which is in the Edexcel Further Pure 2 module.

The use of the method of differences to sum infinite series isn’t often touched upon in A-Level and so I included that in the research part. For the skills section i tried to make most of similar to the ones in the official A-Level text book, apart from Question 4 which is harder than you tend to get for the current FP2 syllabus. It is only harder as I don’t give the function you require to apply the method of differences; the students only need to remember that the function tends have a power that is one order above the terms you are trying to sum and it drops out fairly easily. The stretch questions I have taken from the old Rostock and Chandler books that I love, these are harder than the ones typically seen at FP2, but definitely not insurmountable.

This sheet is available here or in the Algebra section of Jo’s site.

Today I am hostingthe first #mathscpdchat of the new academic year. Between 7pm and 8pm we will be discussing the following topic: “How do you demonstrate to your pupils that you have a personal love of learning maths?”

Since being asked to host this, I’ve been thinkning quite a bit about this and this short post briefly distills some of that thinking.

Of course there are some areas of math that I have a personal leaning towards, but I would say that I love learning about any are of maths. How to demonstrate this to a class at school is somewhat harder to pin down. Rightly or wrongly, teachers are often perceived by their students as all knowing and so as far as they are concerned there is no more learning for us to do – I beleive it is important ot challenge this perception.

Below are some things that I think can help us show a love of learning.

• This is probably easier with an A-Level class, but I will normally choose exercises to do on the board that I haven’t looked at before. This serves two purposes in my opinion;it slows me down so that I don’t gloss over any exposition that may be key to someone else’s understanding and it also demonstrates that I need to try different approaches, or learn a new technique in order to be able to master it.
• Visibly reading books about mathematics. My desk will often have a book that I am currently reading, often a popular maths title (so that if a student asks me about the book I can talk about something fro the book in an accessible way. If I am trying to promote persistance with an A – Leve class, sometimes reading during the lesson can also disuade them from asking me a question too early in addition to showing that I am actively seeking out new knowledge about mathematics.
• If possible talking about current mathematics research that has made the news is good for many other reasons than just showing that I enjoy learning about mathematics. For example, the new result about tiling pentagons is accessible to anyone with a basic understanding of interior angles of a pentagon. 
• Promoting the love of learning of any subject by being interested in discussing any topic or subject with a student. I think the love of learning is infectious and so showing that you enjoy learning and see it as something rewarding in itself has to be a good thing. 
• Being positive when teaching any topic. Students seem to be scarily perceptive of whether we like (or value) a particular topic – I have definitely made this mistake before. 

I’m looking for better ideas than this to demonstrate my love of learning mathematics. It seems very hard(to me at least) to convey the excitement I feel when working through a new problem or learning some cool new result. 

I’m looking forward to discussing it in just over half an hour. Join in tonight’s #mathscpdchat at 7pm (UK time)!