Friday, 18 November 2011

Programming for learning mathematics: Project Euler

Last week I went to two events where the subject of programming came-up, and specifically how useful a tool it is for the learning of mathematics.  Essentially to be able to program a computer to perform a process you need to understand it first and also the process of thinking about how you would construct a program can help you to understand an idea.

Project Euler
At both events Project Euler was mentioned as a great resource/community to encourage people to learn maths through programming.  Michael Borcherds (twitter.com/mike_geogebra) suggested that I might be interested in it at the Computer Based Math summit: www.computerbasedmath.org/.  Then a couple of days later Matt Parker (twitter.com/standupmaths) promoted it at MathsJam: mathsjam.com/.

Project Euler (projecteuler.net/)  is a series of  mathematical/computer programming problems that require some mathematical insight and a little bit of programming knowledge to solve.  However, an understanding of a FOR … NEXT loop and an IF … THEN statement should be enough to get going.  The website is structured so that you aren’t restricted to any particular programming language – you just enter your numerical answer generated by your program and it checks the answer.  It also features a list of the problems you’ve solved and (once you’ve solved a problem) lets you see the forum for that problem.

Programming on the TI-Nspire
I tried the first problem on the TI-Nspire.  The Nspire has TI Basic built into it and is pretty easy to get going on.

One additional advantage is that it has all the mathematical functions built-in and easy to access which is especially useful if you’re using the CAS version.

Python
I tried the second problem using Python.  Python is an open-source programming language that is popular because it is also easy to read.

I downloaded Python from python.org/ and then installed the Ninja IDE front-end from ninja-ide.org/ (and then pointed it at where I’d installed Python).  Ninja is a lot easier to use than the command-line version of Python. 

Programming for learning maths
As generalisation is often the aim in mathematics programming is an excellent tool for learning the subject.  When programming is viewed as explaining a generalisation to a computer it is easy to see why it is so powerful.

Thursday, 3 November 2011

Using an Interactive Whiteboard (IWB) effectively for teaching Maths

Many mathematics classrooms are installed with interactive whiteboards (IWBs); but I often get asked by teachers how they can use them to their full potential.  There is a lot more that can be done with an IWB other than displaying a static demonstration.  The Ofsted report of 2008 highlighted that the full potential of IWBs is not being capitalised on in Maths lessons: “... too often teachers used (IWBs) simply for PowerPoint presentations with no interaction by the pupils.” (see http://www.ofsted.gov.uk/node/2255).  I discussed some of the problems with PowerPoint in my previous post: “What’s wrong with PowerPoint for Teaching Maths”.  In spite of this there are some features of IWBs that make the particularly useful for teaching maths relating to the immediacy of dynamic software.

The immediacy of dynamic software 
When using an IWB the projector is (almost) always on therefore it is easy to use a piece of mathematical software for a small component of a larger lesson.  This is especially useful when there is an object that can be dragged where it is on the screen (as opposed dragging it using a mouse which his hidden from the students’ views).

In comparison in the days before IWBs, when there were only a couple of projectors in the college I was teaching in, I found that if I signed the projector out for a lesson I had a tendency to overuse it – what could have been a very effective 5 minute explanation using some dynamic software turned into a longer and less-effective part of the lesson.

There are many pieces of dynamic software that can be used with an IWB but two that I find particularly effective are Autograph and TI-Nspire.

Autograph
Autograph has an IWB mode that can be enabled from the top menu (shown in the red square in the image below).  This implements a few very useful features:
  • On-screen keyboard (this is a very useful keyboard with built-in mathematical characters that works in other applications)
  • Multiple select without shift
  • The scribble tool: this allows you to make annotations that remain when the software is changed
  • Thicker lines

A single side of A4 with useful IWB tools can be found at: http://mei.org.uk/files/ict/autograph_tools.pdf.  This can be printed and stuck to the wall next to your IWB.

TI-Nspire
Nspire is a fantastic bit of software that I’ve also blogged about previously: “TI-Nspire 3.0”.  One particularly useful feature when using it on a IWB is the Keypad in the Documents Toolbox panel.  This displays an Nspire handheld keypad on the screen which can be used to fully operate the software.  This means that the software can be used without leaving the front of the class and, perhaps more importantly, the students can follow the steps on handhelds.  This is a very easy way to encourage student use of ICT.


Developing your IWB use
Subject-specific IWB professional development is not always easy to find so most teachers will develop their skills through using one.  The process of learning to use an IWB is an important one and it is likely that in the first instance teachers are going to want to use it in the same was as a traditional whiteboard.  This is not something to feel guilty about; however, it is useful when do this to consider how skills can be developed so that over time it is being used to its full potential.

Tuesday, 11 October 2011

The problem with Powerpoint for teaching maths

There are a lot places on the internet where the are negative opinions about Powerpoint.  I don't want to repeat these arguments here, instead I wish highlight particular problems relating to the use of Powerpoint in the teaching and learning of mathematics.  These are that:
  • It reinforces a view of mathematics that it is a series of algorithms to be rote-learned;
  • It can reduce the amount of student-centred use of ICT in learning mathematics;
  • It is usually a static form of mathematics and there are many easy tools for creating equivalent dynamic forms of mathematics.
Reinforcing a limited perception of mathematics
Many Powerpoint presentations for mathematics feature a question with the stages of a solution presented.  This can have a negative impact of students' perceptions of mathematics.  There is a link between students' perception of mathematics and how successful they are.  Students who perceive maths a series of unrelated recipes for solving problems are less successful than those who see it as series of related ideas.  Displaying a single, predetermined method to solve a problem can reinforce the perception that mathematics is about learning the method for each type of question which reinforces the perception of it being about unrelated recipes that need to be rote learned.

Similarly it doesn't allow space for students to ask "what if...?" type questions or to suggest alternative methods for solving a problem or opportunities for linking with other areas of mathematics.  For example if the question is "solve x² + 5x + 6 = 0" and the solution presented is to factorise the students may perceive there is no value to sketching the curve, completing the square or applying the quadratic formula, or possibly, and even worse, that they if they'd tried to solve it using one of these methods that they are "wrong".

Reducing the student-centred use of ICT
The potential that digital technologies, or ICT, can have in the mathematics classroom is widely acknowledged.  However, there is a danger that using Powerpoint as a presentational tool can be seen as fulfilling this requirement and consequently additional, more powerful uses of ICT, such as student-centred use of ICT, may be overlooked. This is missing a huge potential given the impact student-centred use can have on learners' understanding when compared to passively watching a presentation. 

It is easier to produce a dynamic version of mathematics
The mathematics presented in powerpoint is static: if there is a function this cannot be altered easily in the presentation.  By contrast if the function is created in mathematical software it will be easy to alter.  For example, in teaching the relationship between the roots of a quadratic equations and the factorised form, it is straightforward to graph a quadratic function and observe the relationship between the equation and the intersections with the axis.  Not only does this provide a more generalisable demonstration of the relationship, it is easier to do than producing a powerpoint.

Saturday, 10 September 2011

Using the Guardian Data Store for teaching statistics

There are many sites on the internet with data that can be used for teaching statistics but one of the best, and most topical is the Guardian Data Store.  The Guardian Data Store can be found at http://www.guardian.co.uk/data

Raw data on many topical news stories

The data store contains the data associated with stories that are in the news including such varied items as the riots and deprivation, the attendance of MPs and the full data on all the Doctor Who villains.  For many of the items there is a Google Docs spreadsheet of the raw data to download.


Importing the data into software

Whilst it is possible to analyse that data within a Google Docs spreadsheet you can do a lot more by importing it into a statistics package.  Two of the easiest to use are TI-Nspire and Autograph.  With both of these it is very quick to just to copy the data in the spreadsheet and paste it into a list.  All the analysis and the diagrams built-in to these packages can then be applied to the data.


Thursday, 4 August 2011

Using CAS for writing questions

Computer Algebra Systems (CAS) are very powerful tools for mathematics but they are underused in the classroom probably because they aren't allowed in examinations (at least in the English school system).  This lack of use by students means that teachers often overlook how useful they could be for themselves.

One time-saving use for teachers is writing questions with certain properties.  For example if you know you want a cubic with a repeated root you could multiply out appropriate brackets by hand, or you could use CAS to do it:

Similarly you may want a quadratic with specific complex roots:


Most CAS engines also feature calculus tools too so you could use the integration function to find a function with a specific derivative:

In addition to using CAS to write questions teachers can also use to check students' answers.

There are many CAS tools available.  The three that I have used here are:

Wednesday, 25 May 2011

Spreadsheet Algebra

Spreadsheet algebra is a very powerful tool that can be used to teach algebra to students of all abilities and all ages. The major reasons for this are:
  • It is a genuine example of how algebra is used in the outside the mathematics classroom
  • It reinforces the concept of a variable
  • It reduces the likelihood of numerical errors obscuring the underlying mathematics
  • It emphasises the importance of correct syntax
Many students do not see the ‘point’ of algebra. However, they are aware that they may need to use a spreadsheet after they’ve left school and entered employment and so may be more willing to learn mathematics in format that they perceive as more relevant.

Many students do not fully understand the concept of a variable. The use of x as the unknown is alien to many students and can produce misunderstandings. This is not helped by the fact that many students’ first experience of algebra is to solve equations. The result is that they see x as an unknown quantity whose value should be found as opposed to a variable which can be used to define a relationship. A formula in a spreadsheet changes when the variable(s) are changed: this allows students to observe how the output of a function varies as the input varies.  An additional advantage is that instead of typing in the cell-reference when entering a formula you can just click on the cell you want: this makes algebra a physical activity, where you ‘point’ at the variable you want.


Students can often miss the point when investigating mathematical ideas because a numerical error is giving a false result which is obscuring the mathematics. When using a spreadsheet as a tool to investigate mathematics students can rely on the numerical values of calculations and therefore focus their attention on trying to identify and understand any relationships.


The correct syntax for written algebra can be confusing: e.g. you don’t write a multiplication sign; 2 + 3x means multiply by 3 first; etc. Spreadsheet algebra has a slightly different syntax (though helpfully often still uses BIDMAS). This is analogous to learning a foreign language: it will have different grammatical rules, but learning these will improve your understanding of grammar in both of the languages and emphasise why it is important. Learning spreadsheet algebra will improve students’ understanding of the syntax of written algebra.


Examples of tasks
  • Setting up an order form that will calculate total cost when different quantities of products are ordered.
  • Solving equations by trial and improvement.
  • Investigating reverse percentages – what is the cost of an item without VAT?
  • Solving simultaneous equations by trial & improvement and elimination.
  • Setting up a spreadsheet that solves the quadratic equation ax² + bx + c = 0, when the values of a, b and c are entered in separate cells.
  • Setting up a spreadsheet that calculates mean (and standard deviation) from a frequency table.
  • Investigating sequences and series.
  • Investigating exponential growth and decay.
  • Multiplying and finding the inverse of a matrix.
Software
In addition to Microsoft Excel spreadsheets are also available in other software such as Geogebra and TI-Nspire. 

Saturday, 7 May 2011

Euclid – Geometric Constructions (iPhone app)

I’ve been playing with the Euclid iPhone app this week.  It’s a fantastic app and very addictive.  The basic idea is to turn the process of ruler and compass geometric constructions into a game.  You start with some basic constructions, such as midpoint, and as you progress you get set more difficult ones, such as square roots.

The progressive levels of difficulty, as you would expect with a game, works really well and contributes to a sense of achievement when levels are completed!  A particularly nice feature is, as you would expect for something based on Euclid’s Elements, is that when you have completed some of the constructions, such as perpendicular bisector, this then gets added as a tool you can use.


It’s really pleasing to see maths envisaged in a puzzle game format in this way – I’ve certainly learned some geometry from playing it – and would be interesting to know if any teachers have used this with students.  The app can be downloaded from:  http://itunes.apple.com/gb/app/euclid-geometric-constructions/id432735893?mt=8

Wednesday, 13 April 2011

TI-Nspire 3.0

Last Friday saw the launch of the latest version of the TI-Nspire software - version 3.0. This has a few new features in addition to the already excellent version 2:

Adding images
You can now add images to Nspire pages, including as the background to a graphing or geometry page. This is a really powerful tool for relating mathematics to students' experiences outside the classroom.

Differential equations
You can also plot first order differential equations on version 3.0, where the derivative is a function of x and y. It plots a slope field indicating the shape of the general solution and particular solutions can be shown by entering initial conditions (as a single value or list).

3D Graphing
The 3D graphing will plot graphs of the form z=f(x,y). The graphs are displayed really nicely and the window is easy to move. I'm very hopeful that later iterations of version 3 will have the ability to add points and vectors to the 3D graphs so it could be used to for the vectors/3D geometry in A2 Core and A2 Further Pure.

Publish to web and new handhelds
A feature that isn't enabled yet but will be really useful is the ability to publish TI-Nspire files on webpages which can then be viewed (and interacted with!) in any browser. I'm hoping to have some on here as soon as this feature is live. The other major advance is the new handhelds featuring high-resolution colour screens - gone are the days when a pixelated screen meant you couldn't tell the difference between an asymptote and a vertical line and this really brings the technology into the 21st century.

Wednesday, 6 April 2011

Microsoft Mathematics

I’ve recently been exploring Microsoft Mathematics.

Microsoft Mathematics 4.0 is free software that can be downloaded from http://www.microsoft.com/education/products/student/math/.  It has a number of features that are implemented very well in an easy to use format.  The ones that I have used are the in-built CAS, the equation solver and graphing screen.

CAS
The CAS engine appears to be the same as many others – it gives the same results as Maxima for many things.  It’s easier to use than Maxima though: entering and editing are straightforward and there are some useful options, such as differentiation/integration, that are offered immediately upon entering an expression.  Using CAS can make writing questions easier as shown with the example below.


Equation solver
The equation solver will solve any equation, as a normal CAS engine would do, but it also includes the option to display detailed solution steps.  Where there are a couple of usual methods, such as completing the square of the quadratic formula for a quadratic equation, it will display both methods. 


Graphing
Any expression or equation entered in the worksheet can be displayed in the graphing tab.  It automatically recognises parameters so y=x²+bx+c will plot a quadratic with sliders for b and c which can be animated.


Overall this is a very useful piece of software that, whilst it is free, is worth downloading.  The inbuilt triangle-solver and formula-solver in particular will be of use for many teachers and students.

Tuesday, 29 March 2011

Why not "ICT"?

This blog is called “Digital technologies for learning mathematics” and not “ICT for learning mathematics”.  Why the distinction?

The name ‘Information and Communication Technology’ has an impact on how ICT is used in teaching and learning as it implies that the two main uses of technology are for disseminating information and communicating. These two uses of ICT are very important: for example many learners are able to access mathematics through the opportunities for disseminating information and communicating that ICT offers. However, the name Information and Communication Technology can steer teachers down a route where they think ICT’s only roles are for disseminating information and communicating.

Only using technology for disseminating information and communicating overlooks the opportunities that are available, especially using mathematical software, for learners to develop their relational understanding by working in an ICT-environment through exploring, investigating, modelling and programming.  This can result in the overuse of display tools, such as PowerPoint, and miss the huge potential that dynamic software offers.  For example, a dynamic graph show the gradient of a tangent to a point as the point changes is a much more powerful learning tool than a static page about differentiation. 



This issue with the name 'ICT' is also evident in a recent report from NCETM on the use of ICT in mathematics where similar reservations with the name are given and the term “digital technologies” is used instead.  I like the term and have adopted it for this blog (but will probably end up using “ICT” a lot as a shorthand!).   The full report (Mathematics and Digital Technologies: New Beginnings) can be viewed at https://www.ncetm.org.uk/files/3399662/NCETMDigitTechReport2010.pdf