Interactive Mental Mathematics

Recent research has indicated that the teaching of mathematics may be less effective in both England and Scotland than in many European and Asian countries. The third international mathematics and science study (TIMSS), which surveyed the attainment of more than half a million students in more than 40 countries, found that, at age 13, both English and Scottish children scored below the international mean on five of the six reported topic areas (Beaton et al 1996). As can be seen from the table below, which shows a sample of the results, the only area in which British children exceeded the international mean was handling data and probability.

Age 13	Int'nal	Eng	Fra	Hun	Jap	Sco	USA
Overall mean	49	47	51	54	67	44	48
Number	53	48	53	59	71	47	54
Geometry	49	49	58	52	70	46	44
Algebra	44	41	39	52	64	36	44
Data etc.	57	62	63	60	73	58	60
Measurement	45	43	43	49	63	40	36
Proportionality	40	38	38	41	55	34	38

Such findings are of inevitable concern and it is enormously encouraging to see that the Scottish establishment has not just waved its arms dismissively at the problem but attempted to find productive ways of moving forward. For example, it is clear to me as an outsider looking in, that recent events in Scotland - the publication of an informed report followed by appropriate support materials and conferences - have shown how concerned HMI (1999) have been to address this very serious problem.

One way forward, and to its credit HMI have visited classrooms overseas, is to review the literature describing mathematics classrooms around the world to identify the characteristics of, or factors which contribute to, effective teaching. This paper is an attempt to do just that. The review offered has been undertaken within the following loose framework - loose in the sense that informative studies not fitting the criteria have not necessarily been excluded. Thus, in general, this review focuses on studies undertaken in classrooms comprising lower secondary age children and, importantly, studies from countries which formed part of the TIMSS sample in order to provide a contextual, or mathematical attainment, base-line. In reality, and acknowledging the availability of published research, this has meant studies undertaken in France, Hungary, Japan and the United States. The first three because they represent a cross section of countries with relatively high mathematical attainment and the latter because, like England and Scotland, the US appears to have problems of underachievement.

There are dangers in presenting a set of generalities from a limited number of studies undertaken in different national contexts. However, the following, and it is acknowledged that one's perspective on the nature of mathematics and its teaching will inevitably influence choices made, seems a reasonable set of generalities which most teachers of mathematics would recognise and with which many would resonate. Where taught successfully

Much of the above conflicts with the outsider's perspective on British mathematics teaching which has been described as the transmission of skills and techniques. Perhaps more worryingly there seems little in common between the above and a year seven pupil's description of her experiences of a typical mathematics lesson. It is something, despite the passage of time, with which many British teachers and their pupils will be familiar.

'Wait outside if she's not there. Come in if she's there. Sit down. And she tells us what we're gonna do. And she'll probably write up a few examples and notes on the board. Then we'll either get sheets handed out or she'll write up questions on the board. Not very often. We mainly get a textbook. We'll get pages. She'll write up what work to do, page numbers and exercise. And if you finish quick you may get an activity sheet. And that's about what happens' (Clarke, 1984, p.16).

Typical Japanese lessons are based around problems to be solved. Lessons follow a format in which teachers conduct a discussion around the previous lesson's problem before posing the new one. Students work on the problem individually before a public discussion where solutions are demonstrated on the board. The teacher summarises the discussion, students work on a few similar problems and homework is set. The teacher's role appears to be one of presenting interesting problems. He or she offers few explicit verbal instructions in order to allow pupils access to the mathematics irrespective of ability. The more able are expected to find alternative solutions whilst the less able make do with, say, one. Japanese teachers place great emphasis on the planning and delivery of coherent lessons with challenging activities. Lesson plans are highly detailed and attempt to both anticipate and provoke students' thinking. Stigler et al (1996) describe a lesson concerning the area of triangles in which pupils were asked to try to find the best way for calculating the area of a triangle. They were allowed to cut, fold and draw. This phase of the lesson lasted 15 minutes with much discussion occurring. The next phase lasted almost half an hour with nine students offering solutions for discussion and analysis with, importantly, the teacher drawing to the children's attention the essential equivalence of the different solutions. A formula was copied onto the board and for the final five minutes of the lesson students worked through some problems from their text book.

This lesson led into another concerning the equal areas of triangles with equal bases and heights which was followed soon after by one based around the problem shown here.

Two farmers have adjacent fields which look something like that shown alongside.

To make working the fields easier the farmers decide to re-allocate their land in order to make the common border straight. If each farmer is to retain the same area, where should they draw the border?

Two farmers have adjacent fields which look something like that shown alongside.

In France the mathematics offered to children is formal and complex but variously organised to scaffold learning and premised on a belief in the power of discovery. Children engage in investigational or experimental tasks where much emphasis is placed on process and which serve either to establish a relationship which is then subjected to a formal treatment or to demonstrate in a concrete manner something already formally established. All pupils are expected to engage with the formal language, definitions, laws and principles of mathematics; proof and justification; and problem solving. French teachers acknowledge that mathematics is difficult and, unlike their English counterparts, offer their pupils an induction into the complexity of mathematics rather than a reduction from it (Jennings and Dunne, 1996). Lessons comprise several phases. In general they begin by discussing and correcting homework. This is followed by the main phase of the lesson which conforms to the description above before pupils begin a short period of individual working on problems set from a text. Lastly, homework is set. At any time during the lesson pupils may be called to the board to share solutions or to offer ideas because public sharing is seen as an integral part of the process.

In general Hungarian lessons begin with a review of homework set the previous lesson. Solutions, often to one substantial question, are shared publicly with children invited to the board. This may be followed by a brief period of mental work comprising orally presented questions which may revise topics experienced earlier. The teaching phase consists of several episodes in which problems are posed, solutions attempted individually or in pairs, before a public sharing of results involving use of the board and much discussion. At the end of each episode the teacher reviews the outcomes from the task. Finally, homework is set in preparation for the next lesson. Pupils spend little time working through routine exercises or discussing their work in small groups as teachers tend to view the whole class, rather than the individual, as the learner. The problems teachers pose emphasise generality and highlight particular cases as problematic. They facilitate proof strategies and stress links with other topics. Teachers stress abstract ideas although real situations are used to validate the ideas being taught with algebra being presented as the language by which mathematics is communicated. Hungarian teachers appear to offer a mathematically continuous experience incorporating constant revision of ideas covered previously with homework being used both to consolidate ideas and facilitate the transition from one lesson to the next. In short, teachers do not see their task as passers-on of knowledge. Pupils are expected to engage with problems, debate ideas publicly, agree a resolution and, ultimately, an understanding of the issue under consideration. The emphasis lies not in routine practice or applications but in the understanding of mathematical ideas coupled with reasoning and proof. An example of one homework problem I have seen used with a class of twelve year old pupils in Budapest is as follows:

How many isosceles triangles can be found if their areas must be nine units squared and all three vertices lie on grid points of which one must be (3,1)?

A typical lesson in the United States can be divided into two main parts. Firstly, the teacher presents the topic for the day and lectures on it. Secondly, students work individually on practice problems in order to apply and consolidate this new information. American children, like many of their British counterparts, spend long periods of time working independently on practice problems and receive little instruction. There is little opportunity for discussion as teachers see their role as encouraging individual work supported by one to one teacher-pupil conversations. Teachers focus on children's learning of skills and procedures with knowledge being transmitted to an attentive class. Teachers make mathematical content accessible by simplifying it as far as is practicable. They demonstrate ideas rather than allow children to discover them for themselves. Proof and justification, even in an informal sense, are rarely introduced or considered by any participant - the discourse is dominated by low level questions, which teachers frequently answer themselves, and information giving. American classrooms are unreflective environments in which pupils are given few opportunities to think and where little emphasis is placed on performance. Teachers see little need for coherence across lessons and appear unconcerned, at least from the planning perspective, with what might come before or after any given lesson. Lessons may involve a multiplicity and diversity of topics and activities more geared towards appeasement than effective learning.

It would be naive to assume that the above comprises a complete set of factors contributory to mathematical attainment. There will be others which are outside the control of the individual teacher or indicative of particular national idiosyncrasies. Indeed, the mathematics teaching time - both hours in a week and weeks in a year - a child receives varies from system to system. Also, as has been indicated above, the proportions of teacher time given to individual students vary greatly. Indeed, the tradition of the American classroom, and by implication British, whereby teachers spend large times working with individuals or small groups results in many students receiving less teacher time than their Asian or European counterparts. A related issue is the manner in which available lesson time is used. Where attainment is high, not only is little time (never more than a third of a lesson) given to what the Americans call seatwork but also that the quality of the tasks offered to students diminished with increasing amounts of seatwork. The bulk of the lesson time in countries with high attainment seems to be spent on the public discussion and sharing of mathematical ideas. The amount and nature of off-task activities found in classrooms varies considerably. Evidence shows that in both England and the United States, around one fifth of all lesson time is lost through sanctioned and unsanctioned off-task activity with teacher lateness being the major contributor. In many countries, teachers move their lessons into new phases at exactly the same time as British teachers draw theirs to a close.

There is evidence that where attainment is low, as in the United States or Britain, teachers privilege pupils' self-esteem above learning. In Japan and Hungary teachers emphasise learning and, whilst not seeking to embarrass their students, pay little overt attention to such issues. That is, the evidence seems to suggest that a societal emphasis on the individual's self-esteem appears to be a negative indicator of that system's educational attainment - the TIMSS findings in this regard are particularly interesting.

In summary it would seem that there are several generalisable conditions which underpin the successful teaching and learning of mathematics. However, '... systems of teaching are not easily transported from one culture into another' (Stigler and Hiebert, 1997, p19) because teaching is

'a system composed of tightly connected elements. And the system is rooted in deeply held beliefs about the nature of the subject, the way students learn, and the role of the teacher. Attempts to change individual features are likely to have little effect on the overall system. The changes often get swallowed up or reshaped' (Stigler and Hiebert, 1997, p19).

This does not mean that inertia is the only response because perpetuating the status quo equates to our acceptance of attainment lower than international mediocrity. However, progress will require our addressing some substantial issues. One is that in Britain, unlike those countries described above, there is no single and clearly articulated pedagogic tradition to which all teachers subscribe. Indeed, the evidence indicates a variety of British traditions which is one of the causes of many problems of British education. Fortunately, the recent proposals in Scotland and the newly introduced numeracy strategy in England can be seen as steps towards such an objective. A second concerns the nature of mathematics itself and its curricular justification. Outsiders view British teachers as presenting mathematics as a technique-oriented subject at the expense of both mathematics as a process and mathematics as a cultural artefact. The inside story, as manifested in, for example, national curricular documents, is ambiguous in its attempts to reconcile both mathematical applications and mathematical processes. Consequently, teachers, and therefore their pupils, are uncertain as to why the subject is afforded the status it is and how it relates to the wider aims of education which, interestingly, are also ambiguous. A third is that in countries where mathematics is taught successfully education is acknowledged as the servant of equality of opportunity premised on a belief that all children will attain eventually. In France, for example, all students follow the same curriculum in mixed-ability classes in all subjects up to the age of sixteen. This is premised on a constitutional right of equal access to the curriculum whilst, more generally, there is widely-held Eastern belief that all are educable and that it is effort not ability that determines success.

There are other issues which, in essence, are outside the scope of a paper of this nature. We have not considered, for example, the manner in which pupils are grouped for mathematics; assessment and its impact on teaching and learning; the practice of repeat years or, say, differences between national curricula. However, it is hoped that there is sufficient to provoke you to reflect on what you do in your own classroom, to share those reflections with colleagues and to consider how you might collaborate in inviting all pupils to engage with some real and worthwhile mathematics.

Beaton, A.E., Mullis, I.V.S., Martin, M.O., Gonzalez, E.J., Kelly, D.L. and Smith, T.A. (1996) Mathematics in the middle school years: IEA's third international mathematics and science study (TIMSS), Boston MA, Boston College.

Clarke, D.J. (1984) Secondary mathematics teaching: towards a critical appraisal of current practice, Vinculum, 21 (4), pp. 16-21.

Her Majesty's Inspectors of Schools (HMI) (1999), Standards and quality in secondary schools 1995-99: Mathematics, Edinburgh, The Scottish Office Education and Industry Department.

Jennings, S. and Dunne, R. (1996) A critical appraisal of the National Curriculum by comparison with the French experience, Teaching Mathematics and its Applications, 15 (2), pp. 49-55.

Stigler, J.W. and Hiebert, J. (1997) Understanding and improving classroom mathematics instruction: an overview of the TIMSS video study, Phi Delta Kappan, 79 (1), pp. 14-21.

Stigler, J.W., Fernandez, C. and Yoshida, M. (1996) Traditions of school mathematics in Japanese and American elementary classrooms, in L.P. Steffe, P. Nesher, P. Cobb, G.A. Goldin and B. Greer (eds) Theories of mathematical learning, Mahwah, NJ, Lawrence Erlbaum.

Mathematics education and comparative studies: what can we learn from abroad?