Month: August 2015

Towards the end of the summer term my school ran a sixth form taster session for students in Year 10 to promote the sixth form as an option for them; as part of this I ran a taster session for A Level Further Mathematics.

As they were year 10 students they wouldn’t have done a lot of the algebra required to access many of the topics from Further maths so I decided to base a short session around my hierarchy of numbers picture (as I talked about here) as it gave me an opportunity to introduce both complex numbers and the concept of proof.

The first page of the sheet that I used with the students is shown below and the full version can be downloaded from my website.  

  There is quite a lot of text in this sheet, but I went through it relatively slowly giving plenty of time for them to question and think things through themselves – the session got a good response from the students. Note that I did not necessarily discuss things in the ofder that they appear on the sheet. 

This morning I woke to see this article entitled “Weapons of maths destruction: are calculators killing our ability to work it out in our head?” from The Conversation promoted on my Twitter feed. The article is written by Dr Jeanne Carroll of the College of Education at Victoria University, Melbourne she puts forward the view that calculators are not affecting our mental arithmetic “as much as we would like them too”. She cites a blog post from the NFER in November 2014 (following the banning of calculators from primary mathematics tests) indicating that the apparent increased use of calculators at the primary level hadn’t adversely affected pupils abilities at mental maths. She also includes some quotes from Conrad Wolfram (the brother of Mathematica creator Stephen Wolfram) taken from a TED talk of his – I have decided I will blog separately about this at a later date in the summer. Dr Carroll’s article is well worth a read and provides plenty food for thought.

Personally, I feel that student’s reliance on calculators has adversely affected their ability to perform mental arithmetic calculations, or at the very least, their willingness to actually do calculations themselves. I often see students in my KS3 top sets do things like \(8 \times 4 \) on a calculator, and when I question them, they explain that they “just want to be sure”. A Level students routinely use calculators for simple arithmetic (and for calculating the standard trig ratios) with the justification that “only in C1 can we not use a calculator, so why wouldn’t we?”. Stephen Cavadino (@srcav) has also written along these lines here. I personally fell that, especially at A Level, calculators are not used effectively. They seem to be used purely to do relatively trivial calculations, instead of being used to explore other areas of mathematics, that couldn’t as easily be explored without calculators. In fact as it stands, the A Level mathematics qualifications don’t really reflect the way in which technology is used for mathematics – I am anticipating this to change somewhat with the new specification, and MEI’s Further Pure Mathematics with Technology module is an excellent example of how technology could be used at the KS5 level to inspire future mathematicians.

Anyway, I have digressed from the topic of calculators somewhat…. My view is that it is important for students to still be able to perform mental calculations proficiently but I don’t believe that removing calculators completely would help with this. This opinion is partly based on my observation that generally those students who can use calculators effectively (i.e. those who understand where to put brackets) can perform mental calculations perfectly well when required too. Manan Shah (@shahlock) has also observed this. I think it is a shame that their isn’t a mental maths aspect to GCSEs, as the non calculator paper doesn’t really test the ability to use (and choose the best) mental calculation methods.

I love calculators – you my have seen my excitement on Twitter recently over a new purchase – but since going into teaching I have made a conscious effort to not use them that often. I had become very lazy and my speed at recalling basic times tables etc was definitely not as good as it should be (I now enjoy practicing them with my classes on Times Table Rockstars) and I think it is important that I can model to my students in all years that you do not always need to use a calculator. The ability to calculate mentally is very useful as at the very least you can obtain ball park figures to check calculator based calculations.

I think it is time I stop writing, this post feels very rambley! Sorry!!

It’s a Sunday so it is a very short post from me…

This afternoon I was looking at the Higher specimen papers for the new GCSE maths specification from OCR and I really liked the following question from Paper 4  

The mark scheme lists the following examples as suitable answers to the first part:  

I found this question interesting as I haven’t seen something like this in any of the old style papers and I also read something (unfortunately I can’t remember where) about the staggering amount of teachers who couldn’t answer the question “is 0 even or odd”. 

What do you think of this question?

Negative numbers are one of the topics that I find hard to explain. I think one of the reasons is that for us teachers they just make sense – I certainly can’t remember finding them confusing at school myself.

When I’m talking about them at school I often use the usual analogies – money (debts and credits), temperature, positional analogies on a number line etc. Whilst these seem to help with addition and subtraction of negative numbers, multiplication is harder to explain. Using the debts and credits analogy seems to help explain the multiplication of a negative number by a positive number: for example if I associate a debt with a negative number and I have a debt of 20, then it makes sense that if I have 3 of these debts then my total debt is 60, that is, \(-20 \times 3 = -60\). However this analogy doesn’t seem to help when explaining why \(-20 \times -3 = 60\). I quite often end up talking about position on a number line, and a negative number being a direction to the left (a positive number being a direction to the right) and a negative sign indicating a change of direction – but I really don’t find this satisfactory.

Of course, in reality this property of negative numbers can be explained by saying that negative numbers need to satisfy all the algebraic rules of the positive numbers – associativity, commutativity and the distributive property (plus the existence of a multiplicative inverse, namely 1). And so, considering the following sum where we evaluate what is inside the brackets first:

\( -20 \times (-3 + 3) = -20 \times 0 = 0 \)

But of course we can use the distributive property to also say that

\(\begin{align} 0 &= -20 \times (-3+3) \\ &= (-20 \times -3) + (-20 \times 3) \end{align} \)

Having already convinced ourselves that \(-20 \times 3 =  -60 \), then by virtue of addition \(-20 \times -3 \) must be \(60\). This of course can be generalised.

My only issue is that this doesn’t really seem an ideal way to explain it to someone first coming across negative numbers. I’m really interested to hear how you explain this….

I had intended for a different post to go live today, but it looks like the post that was scheduled didn’t save properly so I have quickly thought of something else to write about. Looks like I will get it posted just in time so it is probably going to be shorter than some…. 

I enjoyed reading yesterday’s #summerblogchallenge post from Christine (@MissNorledge) concerning her top 5 sites for starters. Unfortunately though I was reading it on my iPad and consequently some of the links reminded me of one of my great pet hates. 

The two things that drive me round the bend with my laptop is the seemingly constant need to update the Adobe Flash plugin and the Java plugin: neither of these update in a “clean” way, they take far longer than you would expect and break my workflow. I’ve hated Flash for a long time; when I first had a Mac, Flash  would regularly crash safari because of some unexpected error. Unfortunately I can’t write as eloquently about my dislike of Flash as Steve Jobs did in 2010 in his famous “Thoughts on Flash” but suffice to say that in general I am extremely happy that neither my iPad nor my iPhone support Flash and I think that the Internet would be a better place without it. 

The only niggle is that there are some great maths resources out there that have been written in Flash and Christine’s post highlighted some of them and reminded me of this. Ideally I look for resources that are robust and will work on whatever device I happen to be using. This is especially true of I am intending on letting students access some of them – I know that se of them don’t have a computer at home but do have an iPad so anything incorporating Flash or Java applets is a no-no. 

I appreciate that recoding resources to avoid Flash is a massive undertaking but with the completely open standards of JavaScript and HTML5 I genuinely think it is worth the effort. More and more people are using iPads as their primary computing device, some schools are adopting a one-to-one iPad policy and so using the more modern and open standards is surely a no brainer. 

I’ve made my pentominoes resource using CSS, HTML5, JavaScript and the fabric.js library for this reason. I have a few other small resources in various stages of development and am hoping that I will be able to get them completed over the summer. Any of the Geogebra worksheets that I embed in HTML pages (such as this on conic sections and this investigating graph transformations ) use HTML5 so that they work on all devices. I also recently became aware of the website amathsteacherwrites from Jeff (@jeff2869). On this site (as well as some interesting blog posts) there are some great draggable maths activities, such as this matching graphs activity that work on any device. The wonderful Times Table Rockstars developed by Bruno Reddy (@MrReddyMaths) also works on iOS devices (though I understand that there will be dedicated apps coming out which will no doubt improve the UI experience even further for these devices). 

I hope that, over time, more resources which avoid the use of Flash come out and I will try to do my bit and add to them myself. 

Just a short post today where I thought I would share a resource that I found almost a year ago and have used with multiple classes since. This resource works as a starter, main activity or plenary depending on the class and where in a sequence you want to do it.

This resource is from William Emeny (@Maths_Master) and was posted on his excellent Great Maths Teaching Ideas website. It is a card sort of famous number sequences as shown below

  
The high resolution pdf version can be found on his site here. I think it is great how for each sequence you are matching the name, a pictorial representation, a way to produce the sequence, a fact about the sequence and the first few terms of the sequence. There are six sequences contained in this card sort

1. Even numbers
2. Odd numbers
3. Square numbers
4. Fibonacci numbers
5. Cube numbers
6. Triangle numbers

Thanks again for sharing it!

Apparently it was my first WordPress Anniversary on Saturday so I thought I would quickly re-share my first 3 blog posts:

1. Hello World – A First Blog Post
2. Why Teach
3. Thoughts on the Draft Maths A Level Content

I wasn’t as regular with posting when I first started ……