A CROSS-CURRICULAR APPROACH USING HISTORY IN THE MATHEMATICS CLASSROOM WITH STUDENTS AGED 11-16

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1 A CROSS-CURRICULAR APPROACH USING HISTORY IN THE MATHEMATICS CLASSROOM WITH STUDENTS AGED Peter Ransom Bath Spa University Over the past decade I have developed many lessons that use history in the mathematics classroom. These lessons are cross-curricular using events from history to enthuse students in their learning of mathematics. They show that the mathematics used today had, and continues to have, applications to everyday life. As well as being classroom lessons these episodes form mathematics masterclasses done on Saturday mornings around the UK incorporating Texas Instruments handheld technology. The lessons use old and new technology to provide students with an insight into how they are made and used. Recent developments with the STEM (science, technology, engineering and mathematics) initiative in the UK mean these lessons are even more relevant. The two newest episodes are described here. THE EVOLUATION OF THE EPISODES Back in the 1990s I was introduced to the Royal Institution mathematics masterclasses for gifted and talented 13/14 year old students. These took place (and still do) on Saturday mornings and last for 2½ hours. They are organised by a local group of interested teachers and generally take place at a prestigious place covering schools within travelling distance. They put on a programme of 4 to 8 sessions (though most places do 6) inviting staff at local schools to nominate a couple of students who would benefit from these masterclasses. Members of staff are also invited to attend and without these volunteers the sessions would be difficult to run since they attract between 30 and 60 students each session. These volunteers help when students are involved in the tasks since the sessions involve a lot of interactivity. The topics offered vary considerably and cover a wide range of mathematical topics. As the years progressed I felt that there was plenty of material that was suitable for general classroom work and so by developing the masterclasses I do in my own classroom I was able to trial the work with a wide variety of ages and abilities. In my opinion the work should have relevance to the students, so I always try to include material that covers mathematics primarily and history secondly. However science, engineering and geography also play significant parts and all these are brought together to provide a holistic coherent scenario in which to develop the mathematics. Technology changes daily and in preparing our students for the future we need to use the technology of today as well as that of yesterday so that students become digital natives rather than digital immigrants. In my school and masterclasses I use handheld technology by Texas Instruments since that is what we have in school and it fits the work remarkably well, being portable and now wireless.

2 Sometimes the opportunities arise by chance: a suitable anniversary, an interest or an opportunity can be the spur to spending many hours pursuing threads which gets sewn into a rich tapestry of mathematics. The STEM (science, technology, engineering and mathematics) initiative in the UK is aimed at developing an interest in those subjects since they have shown a drop in numbers at degree level over the years. By tackling these areas with scenarios that show how they were dealt with in the past I believe it gives students enthusiasm for mathematics and provides them with a rich background of general knowledge. The episodes I mention here are the two latest ones that have been trialled with teachers and students in the classroom and then improved after feedback from them and my peers in the last year. Galileo and pendula This was the basis of two one-hour classroom lessons with 15 low achieving 14 year olds. I wanted to introduce some wireless handheld technology (TI-nspires) so that the class could develop their IT skills, so we set up an experiment that involved just string and a mass to act as a pendulum. The students were asked to work in pairs since there was a need for one to record data and the other to time the pendulum swinging. After logging on to the wireless system in the classroom, all students were sent a file to the nspire handhelds. This consisted of a page about Galileo and a bit about his work with pendula, but not the result he found. The advantage to the student (and teacher) is that the work they are doing is relayed wirelessly to the interactive white board and refreshed every 30 seconds so the teacher can see how the students are progressing with the work. There is also the added advantage that any student s work can be made to fill the board, so misconceptions can be identified and corrected at an early stage, or good work can be brought to the notice of the whole class. The materials provided in the room included rulers, string, sticky tape, a few stopwatches and some small bits of metal with holes in that can be used as the pendulum s mass. They then worked in pairs to measure the time taken for 10 oscillations of a pendulum (using a stopwatch or their iphone!) and recorded the time and length of the pendulum, after having discussed what factors they thought affected the time of a swing.

3 Fig 1: Screenshots from the pendulum experiment and analysis They recorded the data collected in a spreadsheet page, using that to calculate the time of one swing. They then used a statistics page to show the data from the spreadsheet. With the low attainers in this class we stopped at that point though higher attainers can experiment with changing the variables on the axes until they find a straight line. Built in menus allow the student to find the equation of the straight line that links the length of the pendulum with the square of the time, then by considering the gradient an approximation for the value of acceleration due to gravity can be found. Preciously I thought this activity was too difficult to do without the technology since the low attainers did not have the graphing skills necessary and this would have frustrated their learning. Using the technology removed this blockage and allowed them to realise that there was a connection between the length of the pendulum and the time taken for a single oscillation. The other benefits to this class learning included the following: working cooperatively; developing their creative skills and recording the lengths in metres when they tended to use centimetres to measure. Watching them talk to each other about how and where to put the pendulum was fascinating. Some students attached the string to the doorframe, some to the backs of chairs which they put on the table so they could work with a pendulum length of about one metre. Others tried small lengths of less than 0.2 metres, but soon realised that there were some timing difficulties. This STEM based activity links the science of pendula with the mathematics and involves students thinking for themselves about making working pendula. This episode was repeated with year old students in a large hall at a London comprehensive school in November It was less successful than in the classroom due to other mathematical activities taking place at the same time meaning that the whole group could not be engaged in discussion about the findings. Feedback from students in the final stages was positive however they enjoyed working on group activities more than working as individuals since they had learnt from each other and the new technology had not impeded their learning. (Sometimes it is the teacher who fears that technology they do not understand will interfere with their students learning, but this is rarely the case.) They mentioned that they saw a reason why they had been taught gradient in the past because it now referred to a physical problem.

4 Did the historical aspect make a difference? I have no evidence to say yes or no at this point since no question was directed at this aspect. However the experiment lesson was preceded by one where I asked the students to prepare a PowerPoint presentation about Galileo, his life and his mathematical works. This exceeded my expectations since the students were totally engrossed in the work, demonstrating their ability to use the internet to obtain facts and create impressive files. When trialling new work with my students I never know what will happen, but with over 30 years experience of the year classroom I think I can judge reasonably well what will be accepted. In the plenary at the end of a session students are asked about what they learnt and it is only the comments that students offer at the end of the lesson that give me an insight into whether these lessons are successful or not. Comments such as I learnt loads today about Galileo and some maths ; It was good working in pairs encourage me to continue with such lessons adapting them for different classes. Comments such as It was too hard ; We didn t have a stopwatch means I make amendments to the work and try again with another class. Comments such as That was a waste of time (!); We didn t understand anything means I abandon the work and try a different approach. This episode survived and will be developed further over the next year. Brunel: building bridges Isambard Kingdom Brunel ( ) was an engineer who designed and built many things thought impossible in his day. His Clifton Suspension bridge still spans the Avon gorge near Bristol. Much has been done in the UK to encourage teachers and students to work with this image (and others) using dynamic geometry software to find the equation of the bridge s curve. Fig 2: Isambard Kingdom Brunel yesterday and today I decided to make this a more practical activity by investigating the shape of the curve caused by a suspended chain. Initially the work was trialled with prospective teachers at Bath Spa University, then with actual teachers at the Brunel Museum in Rotherhithe, London. The work with 14/15 year old students was on a large scale with 90 of them

5 in a school hall. The equipment used was a chain (1 metre long) and framework I have previously used in another activity. It is possible to use a cereal box with the front panel cut away to suspend the chain. Fig 3: The chain and framework They also work in a more practical sense by suspending chains on a framework and measuring the horizontal and vertical distances from an origin. This data they put into a spreadsheet page on a TI-nspire handheld then try to find the equation of best fit using either a graph or statistics page. This helps develop their measuring, algebraic and IT skills. Fig 4: Screenshots of the data and successive transformations of y = x 2

6 Fig 5: Screenshots of transformations of y = cosh x Teachers and students appreciated the practical approach and felt that it strengthened their understanding of transformations. The work with the hyperbolic trigonometric function was not mentioned to students under 16 since that is not part of their syllabus, but the link between the transformations of the quadratic and coshx was appreciated by the teachers, though that aspect has yet to trialled in the classroom. It is possible to visit the Brunel collection at Bristol University and I have acquired photocopies of Brunel s actual calculation books. Some of the mathematics he used is appropriate in today s classroom and worksheets have been developed that have been trialled in various situations. The following illustration is one that involves the use of simultaneous equations. Fig 6: Brunel s calculation of the distances between holes in a rack

7 Students have checked Brunel s solution and commented on the results and looked at the proportion of the two sets of answers. One student found this work fascinating commenting that it was very interesting to see how simultaneous equations were used in real life. Fig 7: Brunel s use of Pythagoras Theorem and calculus Students under 16 work with the Pythagoras manuscript, those who have experience of calculus also work with the Thorney Broad sheet. I cover up some of the terms and solutions and students have to work these out themselves. Fig 8: Brunel s box of mathematical instruments

8 Proportional dividers, seen here at the bottom of the box (note Isambard Kingdom Brunel s initials, IKB, are engraved on them) are used to enlarge diagrams with given scale factors, or used to divide circumferences of circles into a given number of parts. Traditionally the sides are labelled with circles (to divide a circumference into 6, 7, 8..., 20 parts; lines (to enlarge a given line with a scale factor from 1 to 10); planes (to enlarge a given area with a scale factor from 1 to 10); solids (to enlarge a given volume with a scale factor from 1 to 10). The two legs are equal in length and the central slide allows the pivot to move, thus in effect creating two similar triangles. I have produced replicas in card that students cut out and make, using a paper fastener to act as a pivot. This gives a practical use of ratio and develops an understanding of similar triangles, enlargement and dividing a line in a given ratio. The accuracy is quite astounding! Fig 9: Actual and student made proportional dividers Conclusion My experience with using history in the mathematics classroom is in relating the mathematics students do with episodes from history when it was used. I believe students need to see where mathematics has been used to appreciate its importance. These activities were developed for use in the classroom and masterclasses, not as a research project, so the evidence of their success (or failure) is mainly ephemeral through comments received by participants after each session. REFERENCES Brunel, I.K.B. (1838?). Clifton Suspension Bridge volume 2 [book of manuscript calculations by the author]. Brunel Special Collection (CSB Vol. II: calculations, notes & sketches. 1838?-1850). University of Bristol Library Byrne, E. & Gurr, S. (2006). Isambard Kingdom Brunel, a graphic biography. Bristol: Brunel200