Lessons we have learnt: effective technologies for effective mathematics
Institute of Education, University of London
Childrens and adults mathematical knowledge frequently appears to be in a state of crisis a crisis of skills or a crisis of creativity. In UK and USA, there are now waves of enthusiasm for basic skills, mental arithmetic, and target setting. A huge multi-million pound National Numeracy project is now underway in UK and we await the final publication of our Government's Numeracy Task Force. In its preliminary report, (Numeracy Matters, 1998), the TIMSS studies (see for example, Harris, Keys & Fernandes, 1997) were cited as one reason for this new focus:
Studies comparing England's performance in mathematics with other countries show this country to be performing relatively poorly in comparison with others. For example, evidence from the Third International Mathematics and Science Survey (TIMSS) indicates that our Year 5 pupils (aged 9 and 10) are amongst the lowest performers in key areas of number out of nine countries with similar social and cultural backgrounds.' (P. 8)
At the same time, the news from the Pacific Rim reports rather different pressures for change. For example, Lew, (in press) describes Korea, a country which scores very highly on most international comparisons of mathematics attainment, as being in 'total crisis' in mathematics. He illustrates graphically how most students seem quite unable to relate their well-developed manipulative skills to the real world. Lew argues that 'the direction of the mathematics curriculum in Korea should change from emphasis on computational skills and the 'snapshot' application of fragmentary knowledge to emphasis on problem-solving and thinking abilities'. Similarly, Lin and Tsao present a picture of test obsession in Taiwan where college entrance examinations dominate students' (and parents') lives (Lin and Tsao, in press). Both of these countries are planning to encourage more 'open' curricula to include opportunities for mathematical creativity: that is, adapt their curricula to be more like those now being vilified in UK and USA!
Other data from TIMSS suggest that English children are comparatively successful at applying mathematical procedures to solve practical problems and are generally positive about mathematics. Is it possible to retain these strengths while at the same time consolidating arithmetic skills and developing the ability to construct rigorous and systematic arguments? (The latter area is one in which we have shown out students to be surprisingly weak see Healy and Hoyles, 1998). From a UK perspective, we need to try to redefine our curriculum in such a way that builds on the wealth of informal mathematical knowledge students bring to school, while at the same time drawing their attention to mathematical structures and properties and introducing them more systematically to mathematical vocabulary. The mathematical curriculum of the next millennium should harness children's motivation without losing their mathematics and we envisage that the computer might offer just the context to help us to do this but this will depend on the role that computers will play a role in the UK mathematics curriculum which is yet to be decided.
A ROLE FOR THE COMPUTER
It has become part of accepted wisdom in educational circles that the computer by itself cannot fundamentally change either what is learned or how, and that issues of learning and teaching are dependent on more than the simple presence of the computer in the learning situation. This does not prevent policy makers however, from viewing the mere provision of hardware as a determinant of educational change. Recently UK newspapers articles have declared that: 'Laptops will cut teacher workload' Times Educational Supplement, April 24, 1998; 'Internet will mean fewer teachers', Guardian, May 21, 1998. Paradoxically, others promise teachers that they will not have to change at all: 'You don't need to change what you teach or what they learn' (advertisement for Research Machines, a major supplier of computers in UK schools).
Suggesting that technology is an independent agent in change misses the uniqueness of what the computer has to offer; it ignores the dialectic between how cultures structure technology and how technology can shape the culture, indeed the mathematics, which it seeks to model. Of course, if our aim is to deliver prescribed content packages and improved scores in 'closed' tests, then the computer can indeed play a role, and one that is no doubt efficient. There is a major fault line running through computer software for school mathematics, and on one side, lies software designed to deliver existing mathematics curricula, to repackage mathematical knowledge (often with appropriate 'edutainment') and present it in acceptably-wrapped fragments for educational consumption. The tools are closed to the user, who 'answers and receives feedback'.
On the other side, however, lie computational applications which point towards new, more learnable, more widely accessible mathematics; towards a redefinition of what school mathematics might become and who might become involved in it and where tools are malleable and principles are visible. It shifts attention away from the 'machine' to the following crucial questions. What 'mathematics' do we want in our schools? How is mathematics shaped by computer tools? What are our aims for mathematics education?
I was inspired in the early 80's by Seymour Papert's radical vision of a mathematics that was playful and accessible, but at the same time rigorous and serious. We dreamed (and still do!) of children actively expressing mathematics in different ways. We wanted children to learn by conjecture, reflection on feedback and debugging, as part of their own meaningful projects that required planning, sustained engagement with mathematical ideas and the bringing together of a range of skills and competencies. Logo was the vehicle or the catalyst for many of us to try to achieve those dreams. In doing this work, our eyes were opened to students' strategies and potentials computer interaction was a window on to possibilities, an environment to illuminate pupil meanings and interpretations (Hoyles 1985, Noss and Hoyles, 1996).
Since that time, we have designed a many microworlds with different 'open' software around different mathematical 'cores'. We have also undertaken more systematic investigation of the nature of the child's activity and how it can be better understood and guided (Hoyles and Noss, 1992). Inevitably the boundary of what is and is not mathematics has been explored (see Papert, 1992): some say that working experimentally with the computer is mathematics, some that it is not, and many are not sure. The software may have changed but the issues have not and the location of this boundary is still a matter of hot dispute, brought even more into focus in an international forum such as this.
If we want to design investigative environments with computers that will challenge and motivate children mathematically, we need software where children have some freedom to express their own ideas, but in ways constrained so as to focus their attention on mathematics bytools that do 'just enough'. Are there lessons to be learned from all the work that has been done with these sorts of environments over several decades? What do we actually know about how children can better learn mathematics with technology?
Mathematics comprises a web of interconnected concepts and representations which must be mastered to achieve proficiency in calculation and comprehension of structures (for elaboration of this theoretical framework, see Noss and Hoyles, 1996). Mathematical meanings derive from connections intra-mathematical connections which link new mathematical knowledge with old, shaping it into a part of the mathematical system; and extra-mathematical meaning derived from contexts and settings which may include the experiential world. Yet how are these meanings to be constructed? How is the learner to make these connections? To what extent can the software tools encourage this process of meaning-making and connection-making?
A critical weakness of many mathematical learning situations has been the gap between action and expression and the lack of connection between different modes of expression. Does technology magnify these problems of fragmentation and lack of connection or help to solve them? Clearly it depends how the technology is used; a lesson certainly worth reiterating! Technology does nothing in and of itself! Over many years, our central research priority has been to find ways to help students build links between seeing, doing and expressing (see for example, Noss, Healy & Hoyles, 1997). We have shown that technology can change pupils' experience of mathematics but with several crucial provisos:
the users of the technology, (teachers and students), must appreciate what they wish to accomplish and how the technology might help them;
the technology must be carefully integrated into the curriculum with due account of progression and not simply added on to it (see Healy and Hoyles, in press), and most crucial of all,
the focus of all the activity is kept unswervingly on mathematical knowledge and not on the hardware or software.
Computers and the curriculum
But what have been the effects of these experimental approaches to school mathematics? Certainly they are exciting in terms of the new horizons to be explored by teachers and students. They do of course raise the thorny issue of the integration of technology into curriculum plans (simply adding it on is counterproductive): when should this be done, how is sequencing affected and of course why? There is also the question of how to build bridges between understandings developed by interaction with software and more conventional mathematical meanings (referred to by Balacheff as computational transposition). There is a new discourse: new objects and relationships to attend to, different things to do and representations to interpret, fresh misconceptions but, crucially, the potential for more engagement with mathematical ideas.
There is one rather graver implication that needs to be addresses. To date, work with computers in mathematics education has largely been concerned with construction and the potential of software to aid the transition from particular to general cases specific instances can be easily varied by direct manipulation or text-based commands and the results seen on the computer screen (see, for example, Laborde and Laborde, 1995). Yet, even if students develop a sense of how certain inputs lead to certain results, there remains the question of how to develop a need to explain, a need to prove, as part of, rather than added on to, this constructive process. In countries like UK, where proof has all but disappeared from the curriculum, this issue must be addressed urgently if we are to avoid limiting the mathematical work for most children by the introduction of computers. If we fail, the majority of our students will simply be subjected to even more convincing empirical argument - for example, using powerful dynamic geometry tools simply to measure, spot patterns, and generate data.
There is an alternative which we are in the process of investigating. We have designed activities where, through computer construction, students during informal, experimental computer-based activity have to attend to mathematical relationships and in so doing are provided with a rationale for their necessity. Thus, the scenario we envisage is one where students construct mathematical objects for themselves on the computer, conjecture about the relationships between them, and check the truth of their conjectures with the tools available. This forms part of a teaching sequence which also includes reflection away from the computer guided by the teacher, and the introduction of mathematical proof as a particular way of expressing one's convictions and communicating them to others. It is in this way, we suggest, that constructing and proving can be brought together in ways simply not possible without an appropriate technology: formal proof is simply be one facet of a proving culture, revitalised by the experimental realism of the computer work, (Balacheff and Kaput, 1996).
Over the last few years, Lulu Healy and I have devised algebra and geometry teaching sequences which follow these criteria. Our activities were developed after analysing students responses to a nationwide paper-and-pencil survey to assess students' conceptions of proving and proof (Healy and Hoyles, 1998). This questionnaire was completed by 2,459 fifteen year-old students of above average mathematical attainment from across England and Wales. Each teaching sequence was designed 'to fit into the curriculum' and to plug at least some of the gaps our survey had revealed in the understandings of our students. Overall 18 students from three schools have worked through the sequences, each of which took nearly 5 hours of classroom contact supplemented by homework.
I will now briefly the two sequences and some snapshots of a student who engaged in them to illustrate the gains that can be made by connecting skills to creative exploration through computer interaction and to point to some potential pitfalls.
making the step to explaining in algebra
Tim was a quiet and diligent student who knew about proof as something that involved verification and explanation, only recognised it in the context of algebra a natural consequence of our curriculum with its emphasis on generalising and explaining number patterns.
In the first algebra session of our teaching sequence, students are introduced to our microworld, Expressor, in which they build 'matchstick' patterns of number sequences by constructing simple programs. They are encouraged to connect their computer constructions with corresponding mathematical properties, and find a general formula for the number sequence explaining why any conjecture is true or false by reference to computer feedback and to the mathematical structures they have constructed.Similar work with more complex number sequences is undertaken in the third session. We have tried similar sets of activities over many years with considerable success (see Noss, Healy and Hoyles, 1997). Figure 1 below shows a typical starting screen where students are asked to make the two given sequences of mathsticks but while doing this find a way to make any sequence as well.
Figure 1: The Opening Screen. The second and seventh terms of the (visual) sequence are shown.
The way that the sequence is constructed can be captured in thehistory box and of course as Expresssor is written in Logo made into a variable procedure. The first of these two steps is illustrated in Figure 2.
Figure 2: Beginning to see a repeated structure in the code
Tim found this work of generalising through programming both engaging and challenging in fact, he described it as the most enjoyable parts of our teaching. He also saw a strong connection between proving and his computer work because it focused his attention on the how he had constructed the sequences:
T I liked the programming stuff - that helped [to write proofs] because it sort of showed how it was constructed so It helped prove because it showed you how they were made... How that construction was made step by step.
In the second session, students are introduced to writing formal algebraic proofs and helped to 'translate' their Logo descriptions of the mathematics structures into algebra. They are also taught how to construct deductive chains of argument; systematically to start from the properties they had used in their constructions and to deduce further properties. Both of these activities are unfamiliar to UK students.
Let me give an example. Students are asked to investigate the properties of the sums of different sets of consecutive numbers. They construct by programming a visual representation of numbers as columns of dots (shown in Figure 3 below). Students can for example move the bottom right dot to the bottom left, see that it would 'even up' the three columns, and convince themselves that the conjecture that the sum of 3 consecutive numbers is divisible by 3 is always true.
Although these moves can be achieved by, for example, using counters, in Expressor, the visual arrangement has a simultaneous 'algebraic' description which is constructed by the children. In Fig 3 a programcol, has been written to generate 6 (n), 7 and 8 columns. The dots can be dragged into columns as with real counters; but as this is done, a recorded 'history' of the actions is stored (see the history box in Figure 3) in the form of fragments of computer program. This code is executable: that is, it can be 'run' to produce the output (or part of the output) which produced it. There is, therefore, a duality between the code and the graphical output of the dots; the action (on the dots) to produce a new visual arrangement and the expression (in the form of pieces of program) are essentially interchangeable and the code is a rigorous description of the student's action to construct a particular image, and her actions are executable as computer programs. A box n is used to store the smallest of the three numbers and our student might see that what is in the box n hardly matters, and therefore that the theorem is independent of the first number.
Figure 3: A typical Expressor screen to explore the sum of 3 consecutive numbers
How did Tim cope with this activity? In his first session, he had been seeking explanations for a general rule in the general symbolic expressions he had constructed (in the form of programs). He constructed his three columns of dots in Expressor and was faced with a screen rather like Figure 3. Then he wrote: .
But, he obtained this equivalence not a result of a manipulating algebra but by reference to our microworld: to 3 columns of length and a 'tail' of 3. He then argued correctly as his proof that the sum of 3 consecutive numbers always had a factor of 3: "if you add 3 to any factor of 3, then it is still a factor of 3" (he used factor instead of multiple throughout!). Tim generalised this method to find factors of sums of different numbers of consecutive numbers always considering columns of dots and a tail, but flexibly using his visual argumentation. For example, to show that it was impossible for the sum of 4 consecutive numbers to have a factor of 4 and so could never add up to 44, he visually moved dots, as he described in Figure 4:
Figure 4: Tim's proof that the sum of 4 consecutive numbers is not divisible by 4
In all his subsequent activities, both on and off the computer , it became clear that Tim had found two, well-connected ways to explain: constructing symbolic code and manipulating visual expressions. His explanations came from linking logical and general arguments with visual representations (columns of dots) and not from algebra, even though he clearly recognised the latter's importance. This gap in his repertoire of skills is well illustrated in his final homework (Figure 5). Tim creatively generalised 'the dots microworld' into thinking of multiplication as a rectangular array of dots, whose rows could be paired off leaving 'one left over'. But, he was still unable to multiply out brackets correctly!
Figure 5: Tim's inductive Proof
Figure 5: Tim's two explanations
We did not address this disjuncture in our teaching sequence. A major effort I believe is needed to build on all that Tim knows, his visual skills and need for explanations and add to it facility with simple algebraic manipulation.
some snapshots from the geometry sequence
I will mention briefly some insights we gained from our teaching sequence in geometry, simply to illustrate further some points raised in the previous sections. This sequence followed a similar pattern to that in algebra. In the first session, students are encouraged to construct simple geometrical objects on the computer with dynamic geometry software, to describe their constructions, connect each with a corresponding mathematical property, and use the computer to explore or reject conjectures. In the second session, students are encouraged to construct familiar geometrical objects (parallelograms, rectangles) on the computer, identify the properties and relations of a geometrical figure that had been used in their constructions and distinguish some properties that might be deduced from those given by exploring with the computer. In much the same way as in algebra, students are also taught at this point to construct logical deductive chains of argument and write formal proofs based on their computer constructions. In the third session, students are faced with more unfamiliar constructions and proofs, which again they can tackle experimentally on the computer.
So how did Tim fare in geometry? Geometry for Tim, as for most of our students, was far more problematic than algebra. He did make some progress in that he learnt to write clear descriptions of his constructions, translate them into given properties and 'see' deduced properties. The computer work helped Tim 'see' relationships and convinced him of their necessity, but the links he could make between constructions and proofs or even explanations were much more tenuous than in algebra. Tim talked very positively about his constructions but was tentative about how it helped him prove -- or even explain.
T Well you could actually see like if they were congruent - you could take however much you were allowed to take and actually make a triangle. If it was congruent then you could tell it was.
C Tell it how?
T Just by seeing.
C And did that help you write your formal proofs?
T Not really -- not explain or the formal stuff But -- well it made it more enjoyable.
Tim found it hard to appreciate and reproduce 'the game' of proving that is, systematically to separate givens from deduced properties and produce reasons for all his steps. He found the language of formal geometry proofs inhibiting it stopped him 'seeing it all'.
The construction task in the third session did however help him to make progress. Tim had to construct a quadrilateral where adjacent angle bisectors were perpendicular and to describe and justify its properties. Tim found this hard, but, after much experimentation and measuring lots of angles, he eventually 'saw' the key relationship two parallel lines but not by 'just seeing them' but by noticing two equal angles and dragging. The important point is that the measurements for Tim were not simply collecting empirical evidence: they were not only part of the conjecture but also and crucially part of his proof. When he talked about say two angles of 44ö, it was clear to us that he was seeing through the numbers to the general case just as he had done in Expressor, so as in algebra, Tom was using the computer interaction to help him to find explanations.
Discussion and conclusion
We designed out teaching sequences and the ways to incorporate computer work on the basis of the strengths and weaknesses we had identified in our survey of conceptions of proving and proof. This was the landscape upon which we could build and will indeed enable us to make generalisations from the case studies. Curricula must seek to build on student strengths in the case of UK on a confidence in conjecturing and arguing and connect these strengths to new dimensions. Students like Tom respond positively to the challenge of attempting more rigorous proof alongside their informal argumentation. especially in the algebra context.
In Expressor, virtual matches and dots are easy to connect with real matches, but unlike their real world counterparts, they connect just as easily to the visual (dynamic) algebra of the system. Of course, just as with real matches and counters, we could not stop students from simply mobilising the tools to formulate specific cases in unsystematic ways, or using their results to construct tables of empirical data from which to spot patterns. The point is that within this medium such behaviour would make much less sense; unlike pencil-and-paper drawing, there is less cognitive load in adopting a systematic approach based on the visual structure and then to exploit therepeat structure of the programming environment, than painstakingly to place each match.
During students' construction there is an explicit appreciation of the relationships that have to be respected,a mathematical model of the situation. The key insight is that parts of this model are built into the fabric of the medium, they are not only available in the mind of the learner. In traditional mathematical pedagogy, there a gap between action and expression which is difficult to bridge. We believe a central challenge for the design of mathematical learning environments is to make visible that which is normally visible only to the mathematical cognoscenti (see Noss and Hoyles, 1996, for further discussion of this issue). In this way, the level of what can be thought and talked about is notched up a rung or two: and ideas can be explored which are located within the world of the particular, the concrete and the manipulable (expressed through mouse clicks, pieces of programs etc.), yet which contain within them the seeds of the general, the abstract, and the virtual.
Progress was not so marked in geometry most likely because UK students have so little background in the even the most elementary building blocks of geometry: for example they are not familiar with even simple relationships such as perpendicular bisector. This makes it hard for them to fully comprehend the construction process. It is also true that we have rather less experience of the direct manipulation interface in geometry and how to help students 'capture' and reflect upon the construction process. Certainly our students needed more experience of the software how they interpreted dragging is a matter we are investigating.
Clearly, not all UK students are like Tim but case studies of his work and those of the other children will provide us with important clues as to how better to integrate technology into our curriculum
Balacheff, N. & Kaput, J. (1996) Computer- Based Learning Environments in Mathematics, in Bishop, A.Clements, K. Keital, C. Kilpatrick, J. & Laborde, C. (eds) International Handbook of Mathematics Education Part 1 pp. 469- 505
Harris, S., Keys, W., & Fernandes, C. (1997). Third International Mathematics and Science Study (TIMSS), Second National Report, Part 1 National Foundation for Educational Research.
Healy, L., & Hoyles, C. (1998) Justifying and Proving in School Mathematics Technical Report , University of London, Institute of Education
Healy, L., & Hoyles, C.(in press) Visual and Symbolic Reasoning in Mathematics: Making Connections with Computers? Mathematical Thinking and Learning
Hoyles, C. (1985). Developing a Context for Logo in School Mathematics. Journal of Mathematical Behaviour, 4(3), pp. 237-256.
Hoyles, C., Morgan, C., & Woodhouse, G. (eds.). (in press). Rethinking the Mathematics Curriculum. London: Falmer Press
Hoyles, C., & Noss, R. (eds.). (1992). Learning Mathematics and Logo. Cambridge, Ma.: MIT Press.
Hoyles, C., & Noss, R. (1992). A pedagogy for mathematical microworlds. Educational Studies in Mathematics, 23(1), pp. 31-57.
Laborde. C., & Laborde, J. M. (1995) What about a Learning Environment where Euclidean Concepts are manipulated with a mouse? In A. diSessa, C. Hoyles, R. Noss, & L. Edwards (eds.). Computers for Exploratory Learning. Springer Verlag.pp. 241-262
Leung, F. (in press).The Traditional Chinese Views of Mathematics and Education: Implications for Mathematics Education in the new millennium, in Hoyles, C., Morgan, C., & Woodhouse, G. (eds.). Rethinking the Mathematics Curriculum. London: Falmer Press
Lew, H-C., (in press) New Goals and Directions for Mathematics Education in Korea, in Hoyles, C., Morgan, C., & Woodhouse, G. (eds.).. Rethinking the Mathematics Curriculum. London: Falmer Press
Lin, F-L., & Tsao, L-C., (in press) EXAM MATHS Re-examined, in Hoyles, C., Morgan, C., & Woodhouse, G. (eds.) Rethinking the Mathematics Curriculum. London: Falmer Press
Numeracy Matters: The Preliminary Report of the Numeracy Task Force (1998) London: Department for Education and Employment.
Noss, R., & Hoyles, C. (1996). Windows on Mathematical Meanings: Learning Cultures and Computers. Dordrecht: Kluwer Academic Publishers.
Noss, R., Healy, L., & Hoyles, C. (1997). The Construction of Mathematical Meanings: Connecting the Visual with the Symbolic. Educational Studies in Mathematics, 33(2), pp. 203-233.
Papert, S. (1980). Mindstorms. Children, computers, and powerful ideas. New York: Basic Books.