# J. Monaghan

Enseignant-chercheur Ã l'UniversitÃ© de Leeds (UK), responsable d'une Ã©tude sur l'Ã©preuve de calcul formel au baccalaurÃ©at international

I was asked to enter into this debate in the light of my involvement with introducing symbolic calculators into the Higher Level Mathematics programme of the International Baccalaureate Organisation (IBO). I was asked to pay particular regard to MichÃ¨le Artigue’s questions. I preface my comments with a brief personal background. I am a mathematics education researcher and my special interests include learning and teaching mathematics with regard to mathematics for 14-21 year olds and using TICE. I note, in a recent debate on developments in France, that «les objectifs de la recherche ne sont pas ceux des enseignants dans leur pratique quotidienne» [Laborde]. I agree with this but I have been drawn into curriculum and assessment matters (apart from my IBO work I have been involved with a team developing a new 14-19 mathematics curriculum and assessment structure in England). |

What I think I have learned, similarly to Laborde, is that this ‘practical’ work, developing large scale curriculum and assessment structures, is very different (maybe more complex) than ell-intentioned people but their jobs create different goals to researchers’ goals, e.g. the production of curriculum and assessment documents (‘specifications’) and precise ‘markschemes’. As my colleague Tom Roper and I wrote in a recent article (Monaghan & Roper, 2007): There are a number of players and artefacts involved in the production of a high stakes examination... Some of these are: communities ...; individuals with specific responsibilities within these communities; artefacts including reports from communities and/or individuals ... There are also relationships between these players and artefacts, e.g. it is clear to us that the DfES has a strong influence on QCA … QCA demands artefacts such as specifications from examination bodies ... (and these specifications are ‘boundary objects’ between different communities).

I assume that similar comments apply in France and a legitimate activity for researchers is to investigate this network of players and artefacts. I now move on to comment on French material I have read.

The main question MichÃ¨le Artigue asks regarding ExpÃ©rimentation d’une Ã©preuve pratique de mathÃ©matiques au baccalaurÃ©at S appears to be «Comment Ã©valeur de telles compÃ©tences?» My first thought is that the competences in the baccalaurÃ©at S document are not clearly described. I have counted nine occurrences of the word «compÃ©tences» and they are mostly ‘vague’, e.g. «compÃ©tences scientifiques», p.3 and «les compÃ©tences dÃ©veloppÃ©es correspondent Ã celles dÃ©veloppÃ©es par les programmes de mathÃ©matiques (compÃ©tences liÃ©es aux diffÃ©rentes parties du programme ou compÃ©tences transversales)», p.5. Calling them ‘vague’ is not really a criticism as it may be useful to leave a precise characterisation until a later time (or never!). Indeed, in my own work with teachers using TICE in mathematics I have sometimes been frustrated by teachers attending to ‘orders from above’ to ensure that students acquire precise competences such as ‘the ability to copy and paste a spreadsheet cell into another cell’ – with the result that classroom work is directed to attending to precise TICE competences rather than to mathematical aims. But their vagueness in the document creates difficulties for me in addressing MichÃ¨le Artigue’s question and I suspect that there are more ideas circulating in French education circles about ‘competences’, than are described in the baccalaurÃ©at S document.

The clearest account of competences in the baccalaurÃ©at S document appears to be «les compÃ©tences des Ã©lÃ¨ves dans l’utilisation des calculatrices et de certains logiciels spÃ©cifiques en mathÃ©matiques», p.4. This is interesting. Although it relates to mathematics it is a TICE not a mathematics competence. In England this matter has been discussed and in Monaghan (2006) I comment on a disagreement in a government working party on the use of TICE in the assessment of 14-19 mathematics:

... distinct disagreement concerned the assessment of ICT skills on mathematics examinations and again ‘pole positions’ were present in the group: those who saw certain ICT skills as mathematical skills and one who defended the thesis that “we use technology but we assess mathematics”. (p.4)

It appears, from my reading of the baccalaurÃ©at S document, that you have adopted the former ‘pole position’. There is not a right or wrong answer to this (it is, in my opinion, a value judgement) but, with respect, have alternatives been discussed?

I think I will reserve further comment on ‘competences’ until a later date, when I have seen what others in this debate have written – I may be better informed then. I now move on to some comments on the the IBO work I have been involved in. These comments are not directly related to issues arising from the baccalaurÃ©at S document but I believe it is what Luc Trouche wants me to comment on.

For several years I have been the ‘external advisor’ to the IBO for the ‘Higher Level’ pilot course which encourages students to use symbolic calculators (TI-89) in their day-to-day learning and allows students to use these calculators in the examination. My role in the IBO has been to comment a little on the curriculum but mainly on the assessment (to comment on and possibly suggest changes to examination questions). The examination that pilot course students take is similar to the normal IBO Higher Level Mathematics examination but some question in the examination are different or are changed a little. Brown & Davies (2002) provides useful background on the introduction of graphic [but not symbolic] calculators into IBO mathematics.

Some background on the IBO that impacts on the pilot course

- The IBO has an international status to ‘protect’ and so must avoid activities that may be seen as ‘radical’. In particular it wants universities around the world to continue to regard IBO qualifications as University Entrance qualifications, i.e. it does not want universities to say “we do not accept that symbolic calculator qualification”.
- All examinations are prepared well in advance, questions are checked by independent groups and detailed ‘markschemes’ are prepared. This is intended to prevent ‘accidents’ and ensure marker reliability but it may curtail ‘creativity’.
- Comparability to ‘by hand’ examinations is viewed as ‘advantageous’.
- Classroom teachers are involved in the groups described in (2) but individual classroom teachers do not design examination questions for their classes. NB The IBO examination thus appear to have some important differences to the tests considered in ther baccalaurÃ©at S document.

In the remainder of my contribution I comment on one thing I have noticed with IBO examination questions which ‘amend’ Higher Level questions – that these amended questions often seem ‘harder’. I will illustrate three ways in which I feel this has been done.

(1) Writing a similar question that requires a more sophisticated solution strategy, e.g.

**Original question**

The graph of y=2x²+4x+7 is translated using the vector (2,-1) . Find the equation of the translated graph, giving your answer in the form y=ax²+bx+c.

**Amended question**

The graph of y=2x²+10x+13 is translated using the vector (h,-1) . Given that only one of the x-intercepts of the translated graph is positive, find the set of values of h.

(2) Writing a totally new (but harder) question in the same topic area, e.g.

**Original question**

Find the antiderivative of e^{2x}sin(x)

**Amended question**

A curve with gradient (1-2x)/(1+x²) , passes through the origin and intersects the x-axis at (k, 0). Find the value of k.

(3) writing an additional (harder) part to the question, e.g.

**Original question** Solve |ln(x+3)|=1 . Give your answer in exact form.

**Amended question**

Let f be a function defined by f(x)=|ln(x+3)| for x > a.

(a) Write down the value of a.

(b) Solve f(x)=1 , giving your answer in exact form.

**A rough hypothesis**

Tool use transforms mathematical actions. New tool use in examinations, be they IBO or baccalaurÃ©at S mathematics, will often result in actions which are more difficult for students to carry out. But my knowledge of baccalaurÃ©at S mathematics is insufficient to determine whether the example provided in the fiche Ã©lÃ¨ve on p.10 falls into this problem.

### References

Brown, R. & Davies, E. W. (2002) ‘The introduction of graphic calculators into assessment in mathematics at the International Baccalaureate organisation; opportunities and challenges’. Teaching Mathematics and its Applications, 21(4), 173-187.

Monaghan, J. (2006) 14-19 Mathematics and ICT: Curriculum and Assessment Issues, Report to QCA. pp.48.

Monaghan, J. & Roper, T. (2007) ‘Introducing more proof into a high stakes examination – towards a research agenda’in Proceedings of the British Society for Research into Learning Mathematics, 27(2).