Categories
Uncategorized

Call for Submissions

I am hosting this months Maths Teachers at Play blog carnival and o would like your submissions please! 

The Maths Teachers at Play blog carnival is organised by Denise Gaskins (on Twitter as @letsplaymath) and every month gets hosted by someone different. This month I have the honour of hosting, and I’m aiming to post the carnival on 30th July.

 I’d love it if you would submit any articles for consideration; you can either do this directly to me or using the form located here

I really want to get a broad range of posts from all over the world and across all age ranges so I am looking forward to anything I receive……

Categories
Uncategorized

Hierarchy of Numbers

I had seen pictures like this before as it is the classic way of representing how the different types of number relate, but I wasn’t totally happy with any of the ones online. 

So I decided to make my own.   

 I used this in a Year 10 Sixth Form taster session. I will post more about this later in the week. 

I created this image in Geogebra because of how easy it is to draw ellipses and use LaTeX symbols. The file is available here

Categories
Uncategorized

Python and Further Maths 1

Because of my numerical analysis background I am very passionate about teaching coding alongside mathematics. In particular, I don’t think the A Level syllabus really does justice to the numerical methods component of the course. As most real life problems have to be solved approximately I think that numerical methods are incredibly important, but that teaching them without coding them up is silly!

I have always found that once I have coded something my understanding of a method has vastly improved. Because of this, if there is time I think it is good to expose A Level mathematicians to a bit of programming.

Python is the ideal language for this as

  1. It is freely available across Windows, Mac OS X and Linux.
  2. The documentation is fantastic,
  3. The syntax is relatively straightforward.
  4. In it’s simplest form the Python interpreter can function as an advanced interactive calculator.
  5. It contains all the functionality of a modern professional program language (i.e. it is not just an academic curiosity)

I believe that number 4 in the above list is one of the big advantages of Python over a traditional compiled language (such as my favourite language Fortran or C++) as this can make the whole concept of learning a programming language for the first time a little bit less terrifying. For students, being able to see almost immediately what the commands they have just typed is very powerful – it must have been awful learning to program with punch cards!

For my first programming workshop with my Year 12 Further Mathematicians I chose to use the IDLE interpreter that comes with Python due to it’s simplicity.

  
With the Python Shell (the rightmost window in the above picture) you can type commands one-by-one and explore Python interactively. Launching the editor window you are able to write programs and then run them with the interpreter. One restriction of Idle with my school’s setup is that I cannot import my own modules – because of this for the next Workshop I will be using PyCharm. 

I gave my students a 6 sided worksheet, with some notes and examples to work through. See below for an example of the type of exercises and the full worksheet is here. 

 
All my students seemed to really enjoy doing a bit of coding, and I was really impressed with how well they got on. I’ve seen 2nd year undergraduates struggle more when they are introduced to Matlab than they were 🙂 

If you fancy having a go working through the sheet, the codes for the exercises are all contained in a tarball which you can download here.

I will write again about the future workshops.

Categories
Uncategorized

Maths Journal Club

At the weekend I suggested the idea of a Twitter Maths Journal Club. MY intention is for this to run along similar lines to the fantastic Twitter Maths Book club (@MathsBookClub) (blog is here).

I have set up a twitter account specifically for this, so please follow @mathjournalclub to stay up to date.

The plan is as follows: every couple of months a journal article will be selected by a poll and we will then have a twitter discussion for an hour one evening, about a month or 3 weeks after the article has been selected. This will be on a day where there isn’t already some kind of mathschat or #mathsTLP taking place.

As a lot of academic articles are pay-wall protected our choice will be a little limited – so either articles that have open access for a particular journal issue, free to access articles or articles where there are high quality pre-prints available on the author’s website.

My intention is to allow people to suggest articles for the poll on the following month, but to get things started here are the articles on the first poll (together with their abstracts)

  • How Ordinary Elimination Became Gaussian Elimination; Joseph F Grcar – Newton, in notes that he would rather not have seen published, described a process for solving simultaneous equations that later authors applied specifically to linear equations. This method — that Euler did not recommend, that Legendre called “ordinary,” and that Gauss called “common” — is now named after Gauss: “Gaussian” elimination. Gauss’s name became associated with elimination through the adoption, by professional computers, of a specialized notation that Gauss devised for his own least squares calculations. The notation allowed elimination to be viewed as a sequence of arithmetic operations that were repeatedly optimized for hand computing and eventually were described by matrices.
  • Contrasts in Mathematical Challenges in A-Level Mathematics and Further Mathematics, and Undergraduate Examinations; Ellie Darlington (Teaching Mathematics and its Applications) – This article describes part of a study which investigated the role of questions in students’ approaches to learning mathematics at the secondary/tertiary interface, focussing on the enculturation of students at the University of Oxford. Use of the Mathematical Assessment Task Hierarchy taxonomy revealed A-level Mathematics and Further Mathematics questions in England and Wales to focus on requiring students to demon- strate a routine use of procedures, whereas those in first-year undergraduate mathematics primarily required students to be able to draw implications, conclusions and to justify their answers and make conjectures.While these findings confirm the need for reforms of examinations at this level, questions must also be raised over the nature of undergraduate mathematics assessment, since it is sometimes possible for students to be awarded a first- class examination mark solely through stating known facts or reproducing something verbatim from lecture notes.
  • A Glimpse into Secondary Students’ Understanding of Functions; Jonathan Brendefur, Gwyneth Hughes & Robert Eley (International Journal for Mathematics Teaching and Learning) – In this article we examine how secondary school students think about functional relationships. More specifically, we examined seven students’ intuitive knowledge in regards to representing two real-world situations with functions. We found students do not tend to represent functional relationships with coordinate graphs even though they are able to do so. Instead, these students tend to represent the physical characteristics of the situation. In addition, we discovered that middle- school students had sophisticated ideas of dependency and covariance. All the students were able to use their models of the situation to generalize and make predictions. These findings suggest that secondary students have the ability to describe covariant and dependent relations and that their models of functions tend to be more intuitive than mathematical – even for the students in algebra II and calculus. Our work suggests a possible framework that begins describing a way of analyzing students’ understanding of functions.
  • Bridging the Divide – Seeing Mathematics in the World Through Dynamic Geometry; Hatice Aydin & John Monaghan (Teaching Mathematics and it’s Application) – InTMA, Oldknow (2009,TEAMAT, 28,180-195) called forways to unlock students’ skills so that they increase learning about the world of mathematics and the objects in the world around them. This article examines one way in which we may unlock the student skills.We are currently exploring the potential for students to ‘see’ mathematics in the real world through ‘marking’ mathematical features of digital images using a dynamic geometry system (GeoGebra). In this article we present, as a partial response to Oldknow, preliminary results.
  • Using Geometric Images of Number to Teach Mental addition and Subtraction, Peter Lacey (Mathematics Teaching) – no abstract available.


It would be great if you would like to get involved, if you do please complete the Poll.

I really hope you want to get involved, I think it could be a great thing to do. Please suggest articles to include for future polls.

Update: Poll closes on 24th July

Categories
Uncategorized

A Few Views on Mastery

A week or so ago Graham Colman (@colmanweb) asked, on Twitter, for a definition and people’s views on Mastery maths. 

Before I started teaching I found it slightly odd that people were talking about the “mastery curriculum” as a new thing…. Surely, no curriculum would have the intention of not building mastery of mathematics in students. My personal experience of school mathematics was entirely on top sets and I definitely did achieve mastery and fluency with the concepts – it was just a bit boring repeating all the topics almost every year until Year 11. I was quite shocked, to be honest, when I started teaching and saw how weak some students were when they came into secondary school and how little progress some students made throughout Key Stage 3. 

Because of this I was quite keen to get involved in developing a mastery based curriculum for Year 7 and 8. My school has been working in conjunction with others in the East Midlands West MathsHub (@EM_mathshub). The layout of our KS3 is similar to that from ARK’s Mathematics Mastery scheme but not totally the same:

 
I am hopeful that this approach will lead to a deeper conceptual understanding in all students and lead to improved results as time can be focussed on extending students knowledge over time and not just repeating things. 

For a definition of mastery in mathematics as asked for by Graham Colman I think I would say something like:

 “A student has developed a mastery in mathematics when they can apply seemingly disparate  techniques and concepts in novel ways  to solve an unseen problem”. 

In a sense this is what a research mathematician is expected to be able to do. 

Can we expect this of school students?

I think we can, within the confines of the school curriculum anyway – geometrical knowledge could be applied to tackle an algebraic problem and vice versa for example. 

Spending more time and delving into topics in more detail allows students to be more critical, giving them a chance to learn how to evaluate strategies and choose the most appropriate given a particular problem. 

For this to be successful though I fundamentally believe that students need to be yet with the basic properties of number – being able to decompose calculations into stages that facilitate calculation, deep knowledge of number bonds and times table facts. Some people argue that these skills aren’t so important now that everyone has a relative good calculator on their phone – I disagree! As well as being quicker than finding a calculator and then keying in a calculation these skills also allow you to mentally check the whether a result is sensible. 

Last month I read Ian Davies (director of curriculum at Mathematics Mastery) post “Mastery – What it is and what it isn’t!” with interest. I liked how he explained why he felt that he had mastered addition, but not integration (in fact can anyone truly master integration?) He also quotes Helen Drury’s definition of mastery of a mathematical concept – the importance of being able to move between different representations stands out.