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SCIENCE TEACHERS TEACHING MATHS USING TECHNOLOGY, ANCIENT AND MODERN.

Thomas McCloughlin

Centre for Science Education St. Patrick’s College

Dublin City University Drumcondra

Dublin 9 Ireland

+353 1 8842092 tom.mccloughlin@spd.dcu.ie

ABSTRACT In Ireland, mathematics is commonly taught by qualified science teachers due to the shortage of mathematics graduates entering the teaching profession. A further complication is that many of the science graduates may be biologists who traditionally have experienced a lesser emphasis on mathematics than the physical sciences. The minimum requirement for teaching mathematics to junior post-primary level is having taken a module in mathematics in at least first year in university, whereas to senior post-primary level in at least second year in university. However, this university level mathematics is unlike that taught in school in depth, breadth and rigor and student teachers may approach the teaching of mathematics at junior level ab initio constrained as they are by their experiences of senior post-primary and university level maths, teachers and institutions. In short, student teachers who qualify to teach maths feel obliged to imitate their strongest role model whether positive or negative. This work outlines how practical hands-on activities were designed using principles based in science education and applied to mathematics education. The technology employed was specific to the teaching and learning of trigonometry, geometry and algebra and ranged from modern information and communication technologies to the use of traditional navigational and surveying equipment. Two further aspects discussed are history of mathematics with particular emphasis on Hellenistic and Classical Greek writers and the students designing and making their own apparatus. INTRODUCTION This research is focused on the student teacher education of the Bachelor of Science course in Science Education, Dublin City University. This course was initially begun as part of the University's initiative to contribute to a wide variety of action plans to halt the decline of numbers taking is the science at a higher level in school and degree courses in University. The course is a four-year degree course that also qualifies the candidate to teach chemistry or physics in secondary school. Thus, the student takes courses in the three sciences and mathematics in first-year. The student also takes physical science in second year and courses in pedagogy after which they subsequently specialize in physics or chemistry.

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I am specifically responsible for a module in the second year that deals with the pedagogy of science and mathematics at the junior level in secondary school. The subject of this paper will be an approach that I have taken over the last four years in dealing with the mathematics pedagogy section of the junior level course. The reason that the students have to do a course in mathematics pedagogy is pragmatic. Science teachers in Ireland are often, if not always, called on to teach mathematics at least to junior level in secondary school. Subjects may be taught to the junior level in secondary schools if it has been studied up to second year at university; whereas a subject to be taught at senior level in secondary school must have been studied in the third and fourth year at university. I am interested in certain questions that this raises. Does a science teacher teach mathematics in a way that is different from trained mathematicians? Science teachers are used to dealing with a good deal of concrete materials and a huge literature has been built up dealing with a constructivist, investigative, hands-on, collaborative, meta-cognitive, or child centered approach to teaching science. Do science teachers use this experience in teaching mathematics? One approach to answering part of this question is to examine the prior experience of maths in the student teacher assuming that prior experience frames or contextualises the current state of action or being:

“We all carry, as teachers and learners, our own bags and baggage of knowledge, beliefs and attitudes

with us into any new situation” (Crook and Briggs, 1991). This may seem a truism, and as such it may be overlooked or felt that such a truth cannot be handled in the context of teacher education. We stress things to know rather than attitudes to inform since the former are thought to be easier to ‘acquire’ and study for in an exam. Crook and Briggs (1991) describe how in analysing teachers responses relating to their prior experience while being interviewed, the teacher paradoxically appeared to have had negative experiences in their own mathematical education. They also reported how that in one piece of research work undertaken for the ‘Cockcroft’ Report (1982), as many as 50% of people approached to be interviewed about mathematics in everyday life refused.

The aim of this work is to explore one approach for improving the teaching and learning of mathematics at junior post-primary level. Much work has been done regarding teachers’ beliefs regarding the teaching and learning of science and their science content knowledge (Hewson et al., 1995; Smith and Lloyd, 1995; Lloyd et al., 1998; Lenton and Turner, 1999; Smith, 1999; Parker and Heywood, 2000; Levitt, 2001). However, little has been done in Ireland regarding science teacher’s beliefs regarding the teaching and learning of mathematics.

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Crook and Briggs (1991) believe that one’s prior experience (and the subsequent retrospective perception of that experience) of mathematics will constribute to the emerging teachers’ views on the teaching and learning of mathematics. Tobias (1978) relates how everyone is exposed to a wide variety of cultural influences, some of which are mythic. Thompson (1984) believes that there is a strong relationship between the conceptions of mathematics held by a teacher and the teaching and learning styles used by that teacher. Because of this, Thompson (1984) is supportive of the idea that the (student) teacher needs to unpack or expose their prior experience so that they can perceive the extent of a particular experience on their own teaching and learning style. Thompson (1984) relates how most research (in the 1970s) on the relationship between the effectiveness of mathematics teachers and their knowledge focussed on the teachers’ knowledge of mathematics. Begle (1972, 1978); Eisenberg (1977); Clark and Peterson (1976); Jackson (1968); Mackay and Marland (1978); Morine-Dershimer and Vallance (1975)(cited in Thompson, 1984) questioned an original assertion that everything a teacher does is rational and reflective: they reported findings that teacher behaviour is mostly instinctive and intuitive. A pessimistic worldview would suggest that we are enslaved or trapped by our past experience since it is this that frames the present. I would suggest that this might be the case until an alternative is experienced (and makes sense) to the student teachers, thus the second element of this paper – the activities – is one alternative way. But a caveat is needed, sometimes we are happy to be in a trap, and we do not want to be set free. METHODOLOGY AND RESULTS

1. Student Teachers’ Personal Biographies Student teachers (n=18) on the Bachelor of Science in Science Education course were asked to write a narrative account of their own experience of mathematics in primary, secondary and tertiary levels. The students were directed to describe both positive and negative experiences and to write in a personal reflective style. They were asked to keep the narrative to two sides of A4 single-spaced word-processed text. Finally, they were asked to summarise their narrative as a time-line. The responses were read and categories of significant instances recorded. A tally was taken of the number of times the instance was mentioned. On completion of this categorisation, categories with only one occurrence were included under the most similar except where this was illogical. The tallies are represented in Figure1.

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0 5 10 15

teachers

concrete materials (abacus, countingblocks)

tables

constraints in tuition at lower levelexam

lack of tuition for high achiever

entry exam to secondary school

number of students in class

teacher's preparation

overall

content other than tables

tests

shock of change of teaching method

there is always a right and wronganswer'

neithergood norbad

badexperience

goodexperience

Figure 1. Tallies of numbers of occurrence of categories in student teachers’ mathematical biographies

It is clear that the most negative experience of this group of teachers was the learning (or failing to learn) the ‘times tables’ (see excerpt 1) followed by the role of the teacher (see excerpt 2) “To this day I can’t multiply or divide automatically and have to employ a certain amount of working out, for example to multiply eight by nine, I have to multiply eight by ten and subtract eight.”

(Student 51152801) Excerpt 1

“Overall I believe that the attitude of the teachers and their expectation of me had the greatest influence on my learning of maths. I believe that a good pupil-teacher

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relationship is necessary to form a basis from which it is easy to relate to each other and in this way to learn from each other. Learning maths is much more enjoyable if a pupil finds the teacher helpful and interesting rather than criticising and condescending.”

(Student 51639421)

Excerpt 2

Crook and Briggs (1991) described as follows the main categories of their research:

• The role of the teacher • The change from primary to secondary school • Personal failure • Gender • Choice of teaching course • Turning points

Of these, gender and choice of teaching course did not arise here, the second not being relevant except that many of the students who had had a bad experience of mathematics in secondary school purported a change in their attitude on entering university due to the role of the teacher. I would further assert that problems associated with a change of school, issues regarding personal failure, gender may also be related the role of the teacher. 2. Activities In response to biographies from previous years, activities were selected that could challenge the traditional model of the mathematics teacher in Ireland, that is one who teaches without concrete materials in teacher directed, teacher-centered, drill-based approach. That is not to say that such a model is wholly inappropriate. A teacher should have a battery of models of teaching and learning at her/his disposal as a realisation of the idea that not all children learn in the same way all of the time. In the following section, some of the activities are outlined and a brief evaluation provided. Figure 2. outlines one possible framework for incorporating the artefact into a teaching sequence (in a mathematics education lecture).

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Figure 2. Framework for the activities 2.1 Construction of a ‘disc’ or equatorial sundial. Sundials are interesting devices for teaching basic astronomy that involve simple principles of navigation. To that end it would seem an obvious choice in the teaching of the significance of angles, relative location, latitude. However it was found that students are resistant to employing them as a teaching aid for no other reason than they lack credibility in modern life. They are viewed as (paradoxically) unscientific, inaccurate and archaic. It came as a surprise to the students to learn of the geometry involved in the manufacture and use of this type of sundial (Figure 3.). This activity sought to challenge this misconception. This model of sundial was chosen since it demonstrates the importance of latitude as an angle in a physical sense.

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Figure 3. Plan of an equatorial sun-dial from Mayall & Mayall (1994) 2.2 Construction of a clinometer (Figure 4.) Many teachers, and their students are frustrated by lack of concrete materials in teaching mathematics, even for simple equipment like protractors. The notion of making equipment is supposedly abstract This activity was posed as a problem. The students were shown a clinometer and they had an opportunity to handle, feel and use it. They were directed to note the parts and their function: “how could you manufacture the protractor component?”

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Figure 4. Diagram of a home-made clinometer: 30cm ruler, card / acetate protractor, string, plasticene, pin. The protractor was made from the template below (Figure 5.). The students were then sent off outside to find the height of certain buildings on the campus. What other measurement would they have to find out?

Figure 5. Template of protractor. The student teachers were surprised with the simplicity of making a protractor (something which children rarely have when they are needed!) and the clinometer. 2.3 Construction of a trundle wheel / opisometer This activity was posed as a problem. How can you make a wheel out of wood or cardboard that will measure the ground in metres? The students found this difficult because of the way the question was posed. When I re-phrased the question as: “how can you make a wheel with a circumference of 1m?” there were no problems then. This demonstrated the importance of contextualisation of mathematics. Students who have learned by rote the formula for the circumference of a circle, wheel or disc can work well in this context but they appear to be unable to ‘transfer’ to everyday contexts.

pin

ruler protractor

plasticene thread

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Figure 6. Home-made trundle wheel made from cardboard or wood and a broomhandle (a hole needs to be drilled at one end and a bolt with washers placed through the handle and wheel. A marker then needs to be placed on the wheel to denote one complete revolution. 2.4 Use of the Sextant The trigonometry section of the junior level course typically uses navigational problems. A ship travels in a certain direction (not a context familiar to many students any longer) and deviations are made and a problem posed. Children have to be taught compass directions which can typically involve the production/design of ‘wet’ or ‘dry’ compasses.

“A ship k is 8km due east of a buoy b. Another ship l is due north of k.

If the ship l is E 40 N of b, Find the distance from k to l in km,

Correct to one decimal place” A brief history of the sextant (Figure 9.) was given to the students, including Steve Callahan’s improvised sextant which helped him survive 76 days in a rubber liferaft as he was making a transatlantic crossing, (Figure 10.) with discussion of advantage sand disadvantages of each with cultural influences of precursors and other instruments:

Broomhandle

Thick cardboard or wooden disc

Marker

Bolt

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Polynesians and the latitude hook; Arabs and the Kamal and Quadrant; Greeks and the Astrolabe and Compass; Chinese and the Compass; etc.. The students were told to set up the room as a shoreline with an ‘observer’ in a ‘boat’. Three ‘readings’ (angles to landmarks on the shore) were taken using the sextant. The observer moved to a second position, emulating sailing along the coast and took three more readings. How far did they travel? How could you work out how fast they were travelling? Would the problem be easier if the height of one of the landmarks were known, or the height above sea-level a landmark was built? (Figure 7.) What if I were lost and not sure where I was in relation to the land? Pick out three landmarks on land. Measure the angle between the centre feature and one of the other features, and note the angle on a piece of paper. As quickly as you can, measure the angle between the centre feature and the third feature. Lay out the three angles on a piece of tracing paper so that the angles have a common centre point. Move the tracing paper around on the chart until the lines are positioned so as to run through the three features. The point of interaction of the three angles is your position.(Figure 8.) This same technique can be used on land to find one’s position accurately.

Figure 7: Finding speed using a sextant

Figure 8: Using the sextant as a Pelorus

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Figure 9: DAVIS mark 3 teaching / emergency sextant as used in these activities

Figure 10. Construction of a ‘pencil’ sextant

(taken from Fisher, 1995)

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Trachtenberg Algorithms This was included in response to the students telling me that they could not recite their times-tables. Lack of knowledge of the times-tables hinders a person in simple arithmetic, however it is late in one’s life in secondary school to have to go back and learn them. An alternative system of arithmetic is that of Jakow Trachtenberg (born 1888). It is important to note that the system does not replace tables. In the long multiplication protocol, tables are needed but the work I have done with children and students concerns ‘short’ multiplication. Cutler and McShane (1978) use the following example for multiplication by nine: 08769 x 9

1. subtract the RH figure of the long number from ten (10 – 9 = 1). This gives the right-hand figure of the answer.

2. subtract each of the following figures in turn from nine and add the neighbour: 9 – 6 = 3 3 + 9 = 12 9 – 7 = 2 2 + 6 = 8 8 + 1 (carried) = 9 9 – 8 = 1 1 + 7 = 8

3. At the last step, subtract one from the neighbour: 8 – 1 = 9 Thus 08769 x 9 = 78921

The History of Mathematics, Science and Technology – the Hellenistic influence

Using the history of a subject in order to teach it is not new and a cursory scan of websites for various third level institutions reveals many course with modules dealing with history e.g. Engineering Department at the University of Patras. However, it does appear to be missing in teacher education courses – student teachers usually have to take modules in the history of education (NB!); however many new syllabi have provisions for the ‘nature of science’. Dimitriadis et al. (2001) proposed a historical approach to the teaching of the nature of science for trainee primary teachers and they found that a complete approach of history, philosophy and methodology of science led not only to the improvement of the teachers’ views in relation to the nature of science, but also to a relatively sufficient appreciation of the content. In spite of this, Abd El Khalick et al. (1998) have questioned such an approach.

The student teachers on the module tend to believe that mathematics (and science) is

fixed and is infallible – there is always a ‘right’ or ‘wrong’ answer: in other words, it has no history, if we take history to be a record of its development. One thread, then of challenging students’ pre-conceptions of mathematics (and science) is to explore the historical development of it as a discipline because many of the protagonists in the development of laws or norms made ‘incorrect’ claims also.

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It is particularly noticeable that this idea of a fallible maths and science is problematic

for the student teacher. They want to know how to teach with an authority that will not be questioned let alone start to question ‘mathematics’ itself.

Short Course in the History of Mathematics (with science and technology)

• The The Great Library of Alexandria (Sagan, 1991) • Ptolemaic world and cosmological views of the ancients – the importance of

relaising that knowledge is fluid and dynamic • Eratosthenes and the circumference of the earth (as described by Sagan (1991) in

the Cosmos television series) the importance of experimentation in mathematics • Aristarchus and the size of the sun and the moon (Figure 11. This the oldest, 10th

century of a collection of early Greek astronomical texts. In this folio, proposition 13 is described concerning the ratio of the diameters of the moon and sun of the line subtending the arc dividing the light and dark portions of the moon in a lunar eclipse)

• The attempt to ‘Square the Circle’ http://www.perseus.tufts.edu/GreekScience/Students/Tim/SquaringCircle.html

• Greek technology: The Antikithyra Mechanism (Figure 13.) http://www.mech.upatras.gr/~chondros/HistofTechnology.html (de Solla Price (1974) and Sfetsos (2003)

• Greek science: alchemy in general (Figure 12.), the Leiden Papyrus

The earliest surviving gear mechanism is the Antikithyra Mechanism which dates from the 1st century B.C.E. It appears to be an astronomical device similar to an Orrory that charted the movement of the sun and the moon.

This artefact provides the initial impetus into a discussion of gears and then leads on to experimenting with gear wheels to teach / learn the concept of ‘ratio’. The activity with gears is taken from the Cognitive Acceleration through Science Education (Adey et al., 1995), project, Activity 6. The ‘CASE’ methodology was worked through in this workshop.

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CONCLUSION Ultimately the aim of this module and the way it is taught is to produce students who will use their own skills to teach mathematics well but yet incorporate observed best practice. What constitutes best practice is open to question and ensuring it is observed or experienced at a deeper level is problematic also. The student teachers expressed an initial reluctance to break out and teach mathematics in a reconstructed and new way than was their experience given the well-documented phenomenon of newly qualified teachers adopting the praxis of an internalised role model. It takes years to develop one’s own authentic praxis and some may not choose to. At the end of the course, the students expressed the view that the notion of transferring their skills in teaching science to the teaching of mathematics was something they wished to explore further (as evidenced by their course evaluation sheets). They also thought the idea novel and had not considered even its possibility. Further work is needed in this area. How the student teacher generates their ‘role constructs’ as a science teacher vis-à-vis mathematics teacher could be examined in more detail using Repertory Grid Analysis. This would also elicit their views on the nature of the two subjects and allow further refinement of the course. Also, how are these role constructs different or similar to those of a fully trained mathematician? How much of the student teacher’s reliance on externally generated role constructs is further constrained by the use of text books which for the most part retain a 19th century format in Ireland needs to be examined further also. Textbooks and the preconceived ideas of publishers, teachers, students and their parents, as to what constitutes a ‘good’ textbook, hold such an authoritative place in education in Ireland that even ‘new’ textbooks are highly conservative. Finally the efficacy of the course of activities in the history of mathematics needs to be tested in a quantifiable manner. Changes in motivation, the ability to think transferring ideas between the science and maths domains, and the ability to adapt existing resources to their own context could be starting points for research. REFERENCES Aristarchus. Proposition 13 Vat. Gr. 204 fol. 116 rect mat06 NS.02 http://sunsite.unc.edu/expo/vatican.exhibit /exhibit/d-matematics/Greek_math2.html last accessed 14th April 2004. Abd El Khalick, F.,Bell, R.L. and Lederman, N.G. (1998) The nature of science and instructional practice: making the unnatural natural. Science Education 82 pp417-436. Adey, P.S., Shayer, M. and Yates, C. (2001) Thinking science: students and teachers’ materials for the CASE intervention. Nelson Thornes, London (3rd edition).

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