Laptops In Lessons
Using Internet access devices in Mathematics lessons
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, for example scientific and graphing calculators, spreadsheets, and interactive web applications or mobile apps, as illustrated in the examples below.
Feedback is a fundamental way in which ICT supports learning in Mathematics. Through feedback, learners notice patterns, see connections, and explore by making mistakes and observing 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. With ICT resources, feedback is quick, 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 a trial-and-improvement approach to solve equations using calculators or a spreadsheet, immediately seeing how a change in input affects the outcome.
Year 7 students used a small software “microworld” on laptops, tablets or graphing calculators to develop problem-solving skills and practise their understanding of coordinates. For example, they explored how changing the values in y = mx + c affects the line, matching targets on a grid.
A group of Year 10 students gained a strong understanding of the connection between the equation of a straight line and its graph by generating many examples, varying parameters and observing the effect. Using graphing calculators or web-based graphing tools made this activity easily accessible without elaborate preparation.
Using still or moving images in a modelling activity extends the way feedback supports learning. Still images can be imported into most dynamic geometry or graphing packages. For example, Year 10 students filmed their own basketball shots in PE, imported frames into software, and fitted quadratic curves to introduce the equations of parabolas.
Such activities suit a wide range of attainment. For example, Year 7 students explored the outcomes of repeatedly applying the square-root button to different inputs and described what they noticed.
Online support can take the form of email dialogue, comments on students’ work through a virtual learning environment or learning management system (for example Microsoft Teams or Google Classroom), and responses from educational websites that give instant, objective feedback on answers.
Y3 students are loving the broken calculator challenge this morning! #STEAM #don'tgiveup #GirlsWhoCode @Transum pic.twitter.com/oCw0B06F6h
— Karen Donnelly (@KarenDonnelly16) June 29, 2017
The speed of computers enables students to produce many examples when exploring mathematical problems. This supports their observation of patterns and the making and justifying of generalisations because they can examine a sufficient range of cases.
Students should be able to explore the effects of varying values and look for invariance and covariance. This involves changing values to explore a situation, including the use of ICT.
Students should 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 movement of a second point. They made conjectures about the geometric relationships between the pairs of points and added construction lines or performed transformations to confirm or refute their thinking using dynamic geometry software.
There are various software packages that allow many examples to be explored quickly so that learners can observe patterns in their results. This process helps them to explain what is happening.
Some Year 8 learners used a simple number grid that could be redrawn by increasing or decreasing a counter. They explored the sums of the numbers within a shaded shape for different grids.
Year 9 students entered values for p and q to work out how each of the values in the cells of a spreadsheet was related. The speed with which examples were generated encouraged them to make and test conjectures. They made generalisations in words, then expressed them using algebra.
Computers enable 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 allows the graph to be drawn directly from the formula and values can be traced. Working through a medium that 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 relationship, or understanding that an intersection between two lines on a graph can represent the solution to a problem.
Using graphing tools on laptops or tablets, 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.
Year 10 students made explicit links between a trial-and-improvement strategy to find the roots of a quadratic equation and the graphical representation of this by plotting a scatter graph of their trials and seeing how the points moved closer to the x axis.
Students can use computers to manipulate diagrams dynamically. This not only supports learning by producing accurate diagrams and graphs, it also encourages students to develop the capability to generate their own mental images. The facility to generate many examples helps students to notice what changes and what remains the same and enables them to formulate and test conjectures.
Students should be able to visualise and work with dynamic images.
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 a previously defined relationship.
The same example can be designed in most dynamic geometry or graphing packages such as GeoGebra or Desmos.
Some Year 9 students explored a circle theorem by constructing an appropriate dynamic figure and used geometrical reasoning to make conclusions which they then presented to others.
Students could use a dynamic number line or a graphing package with sliders to explore when two different functions share the same values of x and y. For example, for which values of x do the functions y = 2x and y = x2 have the same y value?
Year 10 students used software with interactive sliders so that the graph of a function could be adjusted in real time. They observed how changing parameters affected the shape and position of the graph and wrote statements to describe these effects.
A group of Year 10 students manipulated 3D images of cuboids to solve Pythagoras problems in three dimensions by unwrapping nets and constructing the related 2D shapes. This supported them in devising a solution strategy for traditional problems often tackled with paper and pencil.
Computers enable students to work with real data that can be represented in a variety of ways, which supports interpretation and analysis.
Some Year 7 students took 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.
Existing databases provide access to much larger sets of data. For example, some Year 10 students used a database of 1,174 babies to explore the distribution of birth weights and, after creating a range of standard diagrams, estimated the probability of a baby over 10 lb. They then investigated the relationship between gestation length and birth weight.
The database is freely available from
https://www.tsm-resources.com/
Year 9 students planned a walk to match a given distance–time graph and then tested their plan using either a classroom motion sensor or a smartphone app that records motion data, with a whole-class display.
When students design an algorithm to make a computer achieve a particular result, they must express themselves unambiguously and in the correct order. They begin to model particular behaviours or develop a set of rules. This engagement with a formal system sets up the opportunity to develop a mathematical habit of mind and to strengthen 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 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.
A simple online version of Logo (free) that can be accessed instantly is Transum’s Online Logo. Students can also try block-based coding such as Scratch, or text-based coding with Python’s turtle module, to achieve similar goals.
They also needed to use the correct syntax. This is not just a technical matter; it gives students an opportunity to engage in a formal system for a real purpose, which is to draw a shape. In this context students can be creative, pursue their own goals and develop a sense of authorship.
The opportunity to teach a computer arises whenever students use formulae in a spreadsheet. It is important that students are given the opportunity to do this for themselves.
For example, Year 10 students taught a spreadsheet to display the volume of a box in the Maxbox investigation, which assisted their understanding of symbolic representations.
Many modern graphical calculators include a programming facility, often with Python support. For example, some Year 8 students produced a range of programs, from simple ones that carried out basic calculations automatically to ones that generated 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 that involves geometry, algebra and logic.
Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for those learning Mathematics anywhere in the world. What is your experience of using laptops in lessons? Do you have any good ideas or suggestions? Click here to enter your comments.
Callum Arthur,
Tuesday, August 29, 2017