Using Internet access devices in Mathematics lessons
There are six major opportunities for learners to benefit from the use of ICT in Mathematics:
These opportunities may be developed through a wide range of ICT - standard and graphic calculators, spreadsheets, interactive software on CD or online - as illustrated in the examples below.
Feedback is the fundamental way in which ICT supports learning in mathematics. Through feedback learners notice patterns and see connections, and explore, making mistakes and seeing the consequences of their decisions. Other opportunities, including working with dynamic images, exploring data and teaching the computer also support the learner through various forms of feedback. Through ICT resources, feedback is not only quick, but also reliable, non-judgemental and impartial.
Students should be able to take account of feedback and learn from mistakes.
A basic example of learning from feedback with ICT is students adopting an exploratory approach through using trial and improvement to solve equations with calculators or a spreadsheet.
Year 7 students used a small software program (or microworld) running on a handheld device to help develop their problem-solving skills and practise their understanding of co-ordinates. The aim of the program was that they should try to find the rhino hidden in a grid. This example uses an application for Texas Instruments graphical calculators, a free download from http://www.education.ti.com exploring y=mx + c
A group of Year 10 students had the opportunity to obtain a good understanding of the connections between the equation for a straight line and its graph as they were able to generate many examples through varying the values of the constants and observing the effect on the graph. Using graphical calculators made this activity easily accessible to the students without elaborate preparations by the teacher.
The use of still or moving images in a modelling activity extends the way in which feedback supports learning. Still images can be imported into most dynamic geometry packages and some graphing packages. In this example, Year 10 students used the images of their own basketball shots filmed in PE imported into a software package to introduce them to the equations of quadratic functions. This example is included in the Teachers' TV programme Secondary Mathematics - Hard to Teach. (http://www.teachers.tv/video/19119). These students were using The Geometer’s Sketchpad software. (www.keypress.com)
An alternative would be to use the video library within the Coordinates and Graphing tool from The Mathematical Toolkit, a free piece of software that can be downloaded or run from the London Grid for Learning. (www.lgfl.net)
Such activities are applicable to the full range of attainment and level of mathematics. For example, Year 7 students explored the outcomes of using the square root button for a variety of inputs.
Online support for students can take the form of email dialogue, responses to students work through a virtual learning environment and websites.
The speed of computers and calculators enables students to produce many examples when exploring mathematical problems. This supports their observation of patterns and the making and justifying of generalisations based on the facility to look at sufficient cases.
Students should be able to explore the effects of varying values and looking for invariance and covariance. This involves changing values to explore a situation, including the use of ICT.
Students should be able to be aware of the strength of empirical evidence and appreciate the difference between evidence and proof. This includes evidence gathered when using ICT to explore cases.
In a geometric context, Year 7 students dragged a point around the screen and watched the movements of a second point. They made conjectures about the geometric relationships between the pairs of points and added geometric construction lines and performed transformations to confirm or refute their thinking.
There are various software packages which allow many examples to be explored quickly so that learners can observe patterns in their results. This process will help them to explain what is happening.
Some Year 8 learners used a simple number grid which could be redrawn by increasing or decreasing the counter. They explored the sums of the numbers within the shaded shape for different grids.
Year 9 students entered values for p and q to support them to work out how each of the values in the cells of a number grid was related. The speed with which the examples are generated encouraged them to make and test conjectures and they made generalisations in words, and then using algebra.
This example is taken from the National Strategies Using ICT to address 'hard to teach' concepts in mathematics project and can be downloaded from the Standards site.
The computer enables formulae, tables of numbers and graphs to be linked readily. Changing one representation and seeing changes in the others helps students to understand the connections between them. Within a spreadsheet an algebraic formula can be used to generate a table of numbers and this can then be graphed. Alternatively, graphing software or a graphic calculator allows the graph to be drawn directly from the formula and values can be traced. Working through a medium which enables students to switch effortlessly between these representations enhances their conceptual development.
Students should be able to make connections within mathematics. For example, realising that an equation, a table of values and a line on a graph can all represent the same thing or understanding that an intersection between two lines on a graph can represent the solution to a problem.
Using a handheld device, Year 9 students entered data, plotted graphs and matched functions for a given quadratic number sequence based on a growing pattern made from square tiles.
This example, which uses the TI-Nspire handheld is included in the Teachers' TV programme Hard to Teach - Secondary Maths Using ICT. (http://www.teachers.tv/video/29853 )
Year 10 students made explicit links between a trial and improvement strategy to find the roots of a quadratic equation alongside the graphical representation of this by plotting a scattergraph of their trials and seeing how the points got closer to the x-axis.
Students can use computers to manipulate diagrams dynamically. This not only supports learning by producing actual diagrams and graphs, it also encourages students to develop the capability to generate their own mental images. The facility to generate many examples also helps students to notice 'what changes and what remains the same' and enables them to formulate and test their conjectures.
Students should be able to visualise and work with dynamic images.
These Year 7 students used a dynamic number line to understand the meaning of variables. As they dragged the point n along the number line, the position of point a changed, according to the previously defined relationship.
This example used the Number line tool from The Mathematical Toolkit, a free piece of software that can be downloaded or run from the London Grid for Learning. (www.lgfl.net).
The same example can be designed in most dynamic geometry software packages.
Some Year 9 students explored a circle theorem by constructing an appropriate dynamic figure and used geometrical reasoning to make conclusions and reported them to others.
This is shown in the Teachers' TV programme Hard to Teach – Secondary Maths Using ICT.
Students could use a dynamic number line to explore when two different functions share the same values of x and y. For example, for which values of x are the values of y the same for the functions y = 2x and y = x2?
These Year 10 students used a new software development which enabled the graph of a function to be dragged, allowing them to simultaneously observe changes in its parameters.
This example uses TI-Nspire software or handheld.
A group of Year 10 students manipulated 3-D images of cuboids to enable them to solve Pythagoras problems in 3-D by unwrapping nets and constructing the 2-D shapes. This supported them in devising a solution strategy for traditional problems solved by paper and pencil.
This example is shown in the Teachers' TV programme Hard to Teach - Secondary Maths Using ICT.
Computers also enable students to work with real data which can be represented in a variety of ways, which supports interpretation and analysis.
Some Year 7 students had recently taken part in an online survey, one of whose questions required them to decide which endangered species they would most like to save. They were then able to compare and contrast their views with students from another part of the country, using a range of graphs and statistical calculations.
The survey was part of the free Census at School project. http://www.censusatschool.org.uk
Existing databases provide access to much larger sets of data. For example, these Y10 students used a database of 1174 babies to explore the distribution of baby weights, and, after creating a range of standard diagrams, measured the probability of a baby over 10 lbs. They then investigated the relationship between gestation length and baby weight.
The database is freely available from
These Year 9 students planned a 'walk' to match a given distance/time graph and then tested their plan using the motion detector and a whole-class display.
This example of a motion detector being used is shown in the Teachers' TV programme New Maths Technology – In the Classroom.
When students design an algorithm (a set of instructions) to make a computer achieve a particular result, they must express themselves unambiguously and in the correct order. They are beginning to model particular behaviours or develop a set of rules. This engagement with a ‘formal system’ sets up the opportunity for developing a mathematical habit of mind, to develop their skills of algebraic thinking.
'Teaching the computer' encourages students to formalise their mathematical thinking, define conditions, sequence actions and express their ideas clearly. When the computer carries out the instructions it has been given, students need to observe the effect, and if necessary refine and improve the procedure they taught the computer.
For example, when Year 7 students devised a simple Logo program to draw a hexagon, they needed to decide on the sequence of actions carried out by the 'turtle' and the angle turned.
They also needed to use the correct syntax. This is not just a technical matter; it gave the students an opportunity to engage in a 'formal' system for a real purpose - to draw a shape. In this context students were also able to be creative, pursued their own goals and developed a sense of authorship.
The opportunity to 'teach a computer' arises whenever students use formulae in a spreadsheet. It is important, in this context, that students are given the opportunity to do this for themselves.
For example, these Year 10 students taught a spreadsheet to display the volume of a box, in the Maxbox investigation, which assisted their understanding of symbolic representations.
Most graphical calculators have a programming facility which is easily accessible. For example, some Year 8 students produced a range of programs, from simple ones which carried out basic calculations automatically to ones which could generate random numbers or sequences. In doing so they generalised and expressed their ideas in a formal language.
Students could build on the logo procedures for drawing a hexagon to try to tessellate them. By trying out the procedures and 'debugging' them, they will engage with a rich mathematical environment, working with geometry, algebra and logic.
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