EVALUATION and PROGRAM PLANNING

PERGAMON Evaluation and Programming Planning 2 I ( 1998) 1% I41

Alternative approaches to teaching, mathematics

Jo Boaler

learning and assessing

Abstract

In this article I describe two different models of teaching mathematics and the different forms of knowledge that students developed in response to these. Five different assessments were used to assess students in two schools, as part of a longitudinal. ethnographic study of mathematics learning. The outcomes of these assessments are reported alongside the students’ reflections of their knowledge

use. within and outside school. Students at a ‘traditional’ school are shown to have developed an inert knowledge that, they claimed, was of no use to them in the ‘real world’. Students at a school using ‘progressive’ methods are shown to have developed more flexible and useful forms of knowledge that they were able to use in a range of settings. The results of the three-year study are used, both to problematise traditional models of teaching and assessing and to inform the perspective of situated cognition. (’ 1998 Elsevier Science Ltd. All rights reserved.

Kc~wortl.c Situated copition; Pedagogy; Assessment; Problem solving

I. Introduction

There is now a growing awareness that notions of ‘learning transfer’ are insufficient to explain the ways that

individuals make use of knowledge, within and outside of schools (Resnick, 1993). Learning transfer theories (Judd, 190X; Thorndike, 19 I 3) characterise learning as a process that occurs in individual’s minds, independent of that person’s goals, their actions and the social setting. Situated cognition theorists have exposed the simplicity of such notions, partly by showing that individuals rarely make use of knowledge gained in school in ‘real situ- ations’, precisely because their goals, actions and interpretations of the two settings are different. However. as acceptance of the ‘situated’ nature of learning increases amongst academics. more and more English schools are increasing the formalisation of school knowledge.

Opportunities to acknowledge the situated nature of learning are largely disappearing therefore. The reasons for this increase in formalisation, particularly in subjects such as mathematics. are easy to locate. Since the Edu- cation Reform Act (ERA) (1988) schools in England have been subject to the pressures of national tests, school performance tables. compulsory school inspections and the newly created educational marketplace (Gewirtz et al., 1995). In addition, international comparisons of mathematics test performance amongst 27 countries in 1997. placed England in sixteenth position, prompting

SOl4Y- 7189 98 SlY.00 (’ 1998 Elsevier Science Ltd. All rights reserved. PI1:SOl3’) 7189(98300002 0

more overt pressures from government ministers for

schools to return to ‘back to basics’ teaching techniques and to abandon discussion, group work and open activi-

ties, in favour of class teaching and textbook learning.

I intend in this article to problematise both the return to formalised teaching that we are witnessing in the U.K. and the nature of the high-stakes assessments that are

used to evaluate school learning. 1 will question, in par- ticular, the capacity of these assessments to distinguish between useful and inert (Whitehead, 1962) knowledge.

1 will also aim to show that movements to increase the formalisation of school knowledge are likely to decrease

the effectiveness of school students’ capabilities in the job market and in their lives outside of school. My analysis

will draw upon the mathematical learning experiences of students taught in two schools, who I monitored over a three-year period. The differences between the under-

standing of these two groups of students, at the end of three years, were vast and the significance of these

differences extended both to theories of situated cognition

and to policy decisions in schools regarding teaching and assessing (Boaler, 1996: 1997a.b.c.d; 199X).

I. I. Situutecl upition

Theories of situated cognition suggest that individuals do not use the knowledge they gain in school in non-

130 J. Boaler/Eduation und Program Planning 21 (1998 J 129-141

school situations because the context, situation and goals that are formed in relation to the different settings are

different. The simplicity of this idea makes all the more remarkable the fact that it is ignored in most conceptions

of schooling. Indeed theories of ‘learning transfer’ that

have come under increasing scrutiny in recent years,

underpin many of the practices in educational insti- tutions. Learning transfer theories are based upon a

notion that knowledge can be taken from one situation to another when information is learned, links with a new

situation are recognised and information is successfully retrieved from memory. These theories have been attacked, both because there is limited evidence to sup-

port their existence and because the theories are based

upon functionalist assumptions (Lave, 1988). The func-

tionalist influence upon these theories relates to the con-

ception of knowledge as a set of tools, which is stored in

the memory to be taken out and used whenever necessary. These tools are seen as discrete entities, impervious to processes of socialisation or to the environment or con-

text in which they are required. Learning transfer is

characterised in this debate as occurring across unrelated or remotely related situations and the complex inter-

relationship between the activity, persons and setting is

not recognised. Mathematics teaching in England, as in many other

parts of the World, commonly consists of a teacher dem-

onstrating abstract mathematical procedures at the front of the class, followed by students practising the pro-

cedures in short, textbook questions (Schoenfeld, 1985; Office for Standards in Education, 1994). The 1980s marked a slight departure from this practice, prompted,

in part, by the Cockeroft (1982) report which encouraged teachers to take a more varied approach, with some use of mathematical investigations, open-ended work, dis-

cussion and group work. However, the late 1990s have witnessed a reversal of this thinking, with many teachers responding to the pressures of national testing with the

idea that they no longer have time for anything other

than class teaching of mathematical rules, methods and procedures. This transmission model of teaching that is

now more dominant than ever in secondary mathematics classrooms (ages 1 l-18 years) is based implicitly and sometimes explicitly, upon notions of transfer. These notions display themselves in the idea that students may

learn standard procedures in an abstract decon- textualised way, which they will then be able to take out and use when they meet demanding problems that require the use of mathematics.

One of the tangible results of the situated cognition perspective has been a support for apprenticeship learn- ing (Hutchins, 1993; Engestriim, 1993; Keller and Keller, 1993). In apprenticeship situations, individuals do not learn about new ideas, they learn to use new ideas and to adapt, re-formulate and change ideas when necessary. This approach to teaching appears to develop indi-

vidual’s competence in various ‘real world’ settings. Increased awareness of the effectiveness of apprenticeship

learning has led to the development of ‘cognitive appren- ticeship approaches’ to school teaching in which school

students learn new concepts and ideas as part of the

projects on which they are working (see for example the Cognition and Technology Group at Vanderbilt (CTGV), 1990). The two schools in my research study

related to this development in an interesting way. In one of the schools the students learned abstract, decon-

textualised procedures which they practised in textbook questions; in the other school the mathematical approach shared some of the characteristics of apprenticeship learning. In particular, students were introduced to

different methods and procedures as part of authentic

problems they were given to solve and they did not.

unusually, learn about mathematical procedures from

textbooks or schemes. These two approaches could be distinguished in terms of the extent to which they relied

upon models of ‘learning transfer’ or knowledge in prac-

tice (Lave, 1988, 1991, 1993, 1996).

I .2. Research design

In order to monitor and consider the knowledge that students develop in school and the extent to which they

are able to make use of ‘school knowledge’ I conducted three-year case studies of cohorts of students in two Engl-

ish schools. My enquiry was focused upon the math-

ematical learning of the students and comprised a longitudinal study of a ‘Year group’ of students in each school as they went from Year 9 (age 13 years) to Year

11 (age 16 years). I chose the two schools because they offered completely different approaches to the teaching

of mathematics, but the student intakes of the two schools were very similar. In Years 7 and 8 (ages 11 and 12

years) students at both schools had followed the same mathematics approach; this involved working on indi-

vidualised booklets that demonstrated mathematical techniques and required students to answer short ques-

tions practising the techniques. Students worked through the booklets at their own pace, with no teacher ‘input’ from the front. At the beginning of Year 9 the students

at one of the schools changed to a more formal, didactic, textbook model of mathematics teaching whereas the

students at the other school changed to an extremely open, project-based model of teaching. There were no significant differences in mathematical attainment between the cohorts at the two schools at the beginning of Year 9 (x2 test results on grouped data were not sig- nificant). Seventy-five and 77% of students at the two schools attained scores on standardised tests that were below national averages. Both schools were situated in mainly white, working class areas and there were no significant differences in the cohorts of students in terms of sex, ethnicity or social class.

In order to investigate the impact of the different

approaches to mathematics upon the understanding that

students developed, 1 performed ethnographic, case stud-

ies (Eisenhart. 1988) in the two schools, using a variety of qualitative and quantitative research methods. I observed between 80-100 1 hour lessons in each school. I inter-

viewed the teachers at both schools at the beginning and end of the three years, I conducted in-depth interviews with approximately 40 students from each school when

they were in Years 10 and I 1, 1 held conversations with students during lessons about the work they were doing

and I analysed a variety of school documents. In addition

to these methods. which helped me to gain an insight into the day-to-day experiences of the students in their

classrooms, I conducted various different assessments of

the students’ mathematical understanding. These

included short, long, narrow and applied questions and problems that entailed individual. group and practical

work. I also analysed the students’ responses to the national school leaving examination-the General Certificate of Secondary Education (GCSE) which the students took when they were 16 years old.

At ‘Amber Hill’ school there were approximately 200 students in the Year group I followed and the students

were taught mathematics using a traditional. textbook approach. The teachers explained methods and pro-

cedures from the chalk board at the start of lessons.

the students then practised the procedures in textbook questions. In this school the students were grouped according to ‘ability’. Amber Hill was a disciplined school. run by an ‘authoritarian’ (Ball, 1987) head-

teacher. There were many different rules at the school

and the students responded to these with passive behav-

iour and compliance. In mathematics lessons at Amber Hill, students would sit and watch the board in silence

before practising their mathematical procedures in text-

book questions without complaint or disruption. Between lessons they walked in a quiet and orderly way

down the corridors. The students completed a lot of work

in lessons and the teachers moved quickly through the different text books.

JB:

“What do you normally do in a maths lesson?”

John:

“Well sir usually goes over the work we have to do before we do it. so he’ll write it on the board what we have got to do and explain the questions and that and the rules, the basics of what we have to do in the work and then he’ll tell us to get on with it.” (John, Amber Hill. Year 10. Set 1).

The Amber Hill students were motivated and worked

hard, they learned all the mathematical procedures and

rules they were given in lessons and in short, written

tests of their school mathematics they performed well. However, various forms of evidence I collected over the

three years showed that many of the students at Amber

Hill had developed an inert knowledge (Whitehead. 1962) that they were rarely able to use in anything other than

textbook and test situations. The limitations of the form

of knowledge that the students at Amber Hill developed were displayed in their behaviour in lessons. their per-

formance in assessments and their comments in intcr- views.

In lessons at Amber Hill the students worked dutifully

through their textbook exercises. but when they reached

the end of an exercise and needed to move to a different

one. many students would stop and ask for help. The teacher would explain what they had to do and the stu-

dents would continue with their work. In this way. the teachers and the students unintentionally conspired to

restrict the students’ opportunity to think about math- ematical situations. Additionally, if a question required

thought. for example. if it asked something slightly different from the demonstration in the text-book, the

students would try and find non-mathematical cues that

might indicate what they had to do (Boaler. 1996). There is evidence that this sort of behaviour is common amongst

students in the U.S.A. (Schoenfeld. 1985) and the U.K.,

in response to conventional. textbook teaching. Before I move on to show the ways that these inferences were

triangulated by various diRerent assessments and student interviews. I would like to introduce the students from Phoenix Park school.

The approach at ‘Phoenix Park’ school was very

different from that of Amber Hill and the students I followed (approximately 1 IO in the Year group) experi-

enced a model of school mathematics that was extremely

unusual. At Phoenix Park the teachers would introduce an idea or problem to students at the start of a lesson.

the students would then be expected to take the problem and work on it. extend it and adapt it, for approximately three weeks. Students were encouraged to pursue their own lines of interest and change the problems in any way that they wished. The teachers did not teach the students

mathematical procedures in isolation, if students needed to use a piece of mathematics they did not know about,

teachers would teach it to individuals or to small groups within the context of the problem on which they were working. In this way the teaching model shared some characteristics of apprenticeship learning, particularly the learning of mathematical methods within problems that needed to be solved. There were no books or schemes used in the school. until a few weeks before the students

132 J. Boaler/Eualuation and Program Planning 21 (1998) 129-141

took the national GCSE examination. At this time the

school abandoned their use of projects and started prac-

tising more formalised examination questions. At Pho- enix Park the students worked in mixed ability groups.

At Phoenix Park the teachers believed in giving the

students independence and choice. In mathematics les-

sons the students were allowed to take their work to

different rooms and work without supervision as they

were expected to be responsible for their own learning.

The work students completed in class was extremely

‘open’ and students were given no or little structure when they were working on their projects. Discipline was very

‘low-key’ in the school and students were rarely rep-

rimanded in lessons. In any one lesson it was common

for approximately one-third of the students at Phoenix

Park to be wandering around the room or chatting. Work rate was low and the students perceived mathematics to

be a very relaxed lesson.

H:

“It was definitely a lighter lesson-you’d be involved

and if you didn’t want to be involved you’d sort of sit

back and watch it all happen I suppose.” (Hannah,

Phoenix Park, Year 11, JC).

An example of the sort of project that students would be

given at Phoenix Park was to find the maximum area enclosed by a fence built out of 36 gates. Students would investigate this question, taking it in any direction they

wanted to (for example, investigating shape, area, angle;

drawing graphs, tables and charts), for approximately

three weeks. During my three years of work at Phoenix Park it

became clear that the students were developing forms of knowledge that were very different to those of the Amber Hill students and that the Phoenix Park students were

able to combine, adapt and flexibly make use of the mathematics they were learning. This was obvious in

lessons, where the Phoenix Park students were much

more active and independent than the Amber Hill stud- ents. At any one time most of the students would be working on different problems and they could not depend

upon the teacher to help them. As part of their project work they needed to think, plan, adapt mathematical methods and make mathematical decisions. The students

spent large amounts of time ‘off-task’ (Peterson and Swing, 1982) in lessons, but when they were ‘on-task’, they were exercising mathematical thought. They did not receive any opportunities to work procedurally through exercises. This experience led to the students developing more flexible, less inert, forms of mathematical knowl- edge. This was partly due to the students’ experience of using mathematics and partly because of the perceptions about mathematics that they developed as part of that experience.

The important difference between the environments of

the two schools that caused this difference was not related

to standards of teaching but to different approaches, in particular the requirement of the students at the project- based school to think for themselves. When I asked the

students at the two schools to say whether mathematics was more about thinking or memorising, 64% of the

Amber Hill students chose memorising, compared with

35% of the Phoenix Park students (n = 240). At Phoenix

Park the students were not concerned about learning endless rules and procedures, they had been encouraged

to think about what they knew and adapt what they had

learned to fit new and demanding situations. During my three years of work in the two schools I

collected many forms of evidence. The combination and

triangulation of the different forms of qualitative and quantitative data provided many key insights into the

learning and understanding of the students. The lesson observations and student interviews were particularly

informative in this regard. However, I intend, for the next

part of this article, to concentrate upon the students’ performance in a range of assessment situations, both to

provide further evidence for the differences in the forms of knowledge that the students developed and in order to

prompt consideration of the information provided by

different forms of assessment.

2. Mathematical assessments

There were many differences between the experiences of students at Amber Hill and Phoenix Park schools. In

assessments of their mathematical knowledge and under-

standing, broad differences would therefore be expected

between the two sets of students. In order to investigate whether differences existed in the extent, nature or form

of students’ understanding I chose to use a variety of

assessments. These included applied assessments, long-

term learning tests and short contextualised questions. I

also analysed the students’ responses to the national GCSE mathematics examination.

2. I. Applied tasks and related tests

In each of Years 9 and 10 I gave the students an

applied task and a short written test that assessed all the

mathematics I thought they would need to use in the task. In Year 9 1 gave students an ‘architectural task’, in which they were asked to consider a proposed house and decide whether it would pass local authority design rules. They were given a model of the house and a scale plan of the house and they had to look at angles, areas, volumes and proportions. Two weeks prior to the applied task they were given a test which assessed the same areas of math- ematics through short, written questions. To fit in with the school’s teaching arrangements I gave the task and test to students in the top four ability groups at Amber

J. Boaler/Eraluufion arld Program Planning 21 i lYY8) 129-141 133

Table I Volume activity and test results: Amber Hill

Test qu

Activity grade

Correct Incorrect

Correct 23 15

Incorrect 6 9

29 24 ~ ..____. -

38

15

53

Hill (n = 53) and four mixed-ability groups at Phoenix Park (n = 51). Because the students at Amber Hill were

taken from the top half of the school’s ability range and the students at Phoenix Park were not, there was a

disparity in the attainment levels of the samples of stud-

ents. The students in the Amber Hill sample had scored significantly higher grades on their mathematics entry

tests at the beginning of Year 9. However my main aim

was not to compare the overall performance of the stu- dents in the two schools, but rather to compare each

individual’s performance on the applied activity with

their performance on a short written test. The architectural activity comprised two main sections.

In the first part. the students needed to decide whether a

proposed house passed a council rule that said that the volume of the roof of a house must not take up more than 70% of the volume of the main body of the house.

To determine whether the house passed this rule the stu- dents needed to find the volume of the roof and the house

and find the proportion of the roof to the house. In order

to solve this, students could use either the scale plan or

the model. The students’ results for this problem are shown against their test question results in Tables 1 and

2. The activity was graded on a four point scale, but these grades have been collapsed and Tables 1 and 2 show the

numbers of correct and incorrect answers.

Tables I and 2 show that at Amber Hill. 29 students (55%) were successful on the activity, compared with 38

(75%) of Phoenix Park students, despite the fact that the

Amber Hill students were taken from the top half of the school‘s ability range. These tables also show that at

Amber Hill. 15 students (28%) could do the mathematics when it was assessed in a test, but could not use it in the

activity compared with eight students at Phoenix Park

(16%). In addition. 15 students (29%) at Phoenix Park were successful in the activity. despite getting the relevant

Table 2

Volume activity ;md test results: Phoenix Park

Test qu Correct

Incorrect

Activity grade

Correct Incorrect 23 8 31

15 5 20

3x I3 51

Table 3

Angle activity and test results: Amber Hill

Test qu Correct

Incorrect

Activity grade

Correct Incorrect

31 I9 50

3 0 3

34 I9 5.3

test question wrong, compared with six students (11%) at Amber Hill.

In the part of the applied task concerned with angle.

students needed to estimate the angle at the top of the

roof, which was actually 45 . from the plan or the model,

because one of the council rules stated that rooves must not have an angle of less than 70 This was a shorter and

potentially easier task. The results for the roof angle task

and the angle test are given in Tables 3 and 4.

Tables 3 and 4 show that at Amber Hill 50 students

could recognise a 45’ angle in a test. but only 3 1 of these

students could say whether a 45’ angle was more or less than 70’ within the applied activity. At Phoenix Park, 40

out of 48 students who recognised the angle in the test could solve the angle problem. Paradoxically. the least

successful students at Amber Hill were in Set I, the high-

est group. Eight of the 16 students could not solve the roof volume problem and 10 of the 16 students could not

solve the angle problem. In both of these problems this failure emanated from an inappropriate choice of

method. For example, in the angle problem, the IO unsuc-

cessful students all attempted to use trigonometry in order to decide whether the angle of the roof. which was

45 , was more or less than 70 . but they failed to use

the methods correctly. Successful students estimated the angle using their knowledge of the si7e of 90 angles.

Unfortunately. the sight of the word ‘angle’ seemed to

prompt many of the Amber Hill, Set I students to think that trigonometry was required, even though this was clearly inappropriate in the context of the activity. The

students seemed to take the work ‘angle’ as a cue to the method that they were meant to use. Some of the students

gained nonsensical answers from their misuse of trig-

onometry, such as 200 , but they did not appear to realise that the 45 angle of the roof could not have been 200

One year later, when the students were in Year IO, I

gave approximately 100 students in each school another

Table 4

Angle activity and test results: Phoenix Park

Test qu Correct

Incorrect

___~__

Activity grade

Correct Incorrect 40 x 4x

2 I 3 42 9 51

~ ~._____~

134 J. BoulerlE~~aluation and Program Planning 21 il998) 129-141

applied task and short written test and the same pattern of results emerged, but these were more marked. The Phoenix Park students gained significantly higher grades in all aspects of the applied task and their performance on tests and tasks were similar. At Amber Hill the students performed well in the short, written tests but failed to make use of the same areas of mathematics assessed within an applied task.

The performance of the Amber Hill students on vari- ous aspects of the applied tasks given to them in Years 9 and 10 showed that they had difficulty making use of the mathematics they had learned in an applied situation. This did not appear to be due to a lack of mathematical knowledge, but the ways in which the students interpreted the demands of the activity. This will be considered in more depth later in this paper.

2.2. Long-term learning tests

In this assessment students were assessed on a piece of their school work immediately before being taught the work, immediately after completing the work and then six months later. On each of the three assessment occasions the students took exactly the same test. The tests were designed to assess the learning that took place on a particular topic, in a similar style and format to the actual work. Because the Amber Hill students were taught in ability groups, their work was usually targeted at very specific levels of content. This made the design of the assessment questions straightforward. I designed questions that were essentially replicas of the questions they had worked on in their textbooks, with different numbers and contexts. In Phoenix Park the design of the assessment questions was extremely difficult because the students worked on different problems, at different levels of mathematics. The results for the Amber Hill students are therefore more valid than those for the Phoenix Park students. For this reason, 1 will not expand upon this exercise in detail, but I have provided a summary of the results as, even with these limitations, they proved interesting.

At Amber Hill the two groups that were assessed were a Year 9, Set 1 out of nine, and a Year 10 Set 4 out of eight, both taught by the same teacher. Both groups were taught using the typical Amber Hill approach of the teacher explaining methods on the board, followed by the students practising the methods in exercises. The top set group was taught at a relatively fast pace, as was normal for the school. Both groups worked for about three weeks on the topics that were assessed.

The experiences of the two groups at Phoenix Park differed in more fundamental ways. The Year 9 group was working on what the head of mathematics described as ‘the most didactic piece of teaching’ they ever did at the school. This consisted of the teacher showing the students how to do long division, without a calculator, on

the board and then letting them explore division patterns. The work lasted for only two lessons. The Year 10 work, taught by the same teacher, was a more typical Phoenix Park project on statistics that lasted for about three weeks. A summary of the main results is given below:

l The least successful group of the four was the Set 1 group at Amber Hill. This group learned and remem- bered 9% of the work they had been introduced to over a six month period. The Year 10, Set 4 group learned and remembered slightly more of their work-approxi- mately 19%. This was despite the fact that the Amber Hill students were given tests that were exact replicas of their exercise book questions, with similar numbers and contexts. At Phoenix Park the Year 9 group learned and remembered only 16% of their division work over a six month period (but they did only work on this topic for two lessons, rather than three weeks); the only group that was relatively successful was the Year 10, Phoenix Park group, who remembered 36% of the statistics they used during their open-ended projects.

l As the Phoenix Park tests had to be pitched in the middle of the group and some students may not have worked on aspects of mathematics that were assessed, a more realistic comparison of the two schools is pro- vided by the ratio of the proportion of questions answ- ered correctly in the post-test to the proportion answered correctly in the delayed post-test. These pro- portions are displayed in Table 5.

The results of the long-term learning tests show that the Phoenix Park Year 10 students were more successful than students at Amber Hill and the students who worked in a ‘didactic’ way at Phoenix Park. The results of the Amber Hill tests are particularly interesting as the close match between the tests given to students and the work in their textbooks meant that the tests could give a relatively valid assessment of the students’ classwork. These tests showed that the Amber Hill students could remember a reasonable proportion of their work immediately after their lessons, but six months later one group had for- gotten half of what they had learned and the other group had forgotten two-thirds of what they had learned. The Phoenix Park students were significantly more successful at remembering their work over a six-month period.

Table 5

Proportion of items answered correctly in post-test, also answered cor-

rectly in delayed post-test

Amber Hill

Phoenix Park

Year 9

33%

67%

Year 10

50%

83%

The superiority of the students’ performance at Pho-

enix Park in applied mathematical assessments was prob- ably not surprising, given the students’ greater experience of open-ended mathematical activities in lessons. At Amber Hill the students spent the vast majority of their

time working through short, closed exercises assessing

knowledge. rules and procedures that they had been taught by their teachers. Part of the reason that the school

chose to teach in that way was to provide the students

with a good preparation for examinations that assess mathematics in a similar format. This section will present

the results of two different assessments that gave the

Amber Hill students the opportunity to use the math-

ematics they had learned in a more familiar format.

At the end of Year 10 the students were given a set of nine short. written questions set in different contexts.

These questions assessed conservation of number. num-

ber groups. fractions, perimeter and area. On the five

questions which assessed fractions and conservation of number there were no significant differences between the

schools. On the two number group questions the Amber Hill students attained significantly higher grades, mainly

because a large proportion of Phoenix Park students

did not answer these questions. On the two questions involving perimeter and area the Phoenix Park students attained significantly higher grades. Taken overall. the

performance of the two sets of students on these tests was

therefore broadly comparablee this result. which was. in some henses, surprising. was illuminated further by a

consideration ofthe students’ performance in their GCSE

examinations.

When the time came to take GCSE mathematics exam-

inations. the Phoenix Park students faced a number of disadvantages. For example. the students had not met all

of the standard mathematical procedures and algorithms that were assessed in the examination, because they only learned about new mathematical procedures if they hap-

pened to need them during the course of a project. The

students were also disadvantaged in the examination because Phoenix Park did not provide students with

examination equipment such as calculators, because the school could not afford them. The students may also have been disi&antaged by the relaxed atmosphere of the school which meant that few ofthe Phoenix Park students were ‘geared up’ for their GCSE examinations. Indeed. many of the Phoenix Park students reported that they had not bothered with revision for their examination.

The Amber Hill students’ preparation for the exam-

ination was very different. The GCSE examination had

a high profile at Amber Hill and success in the exam-

ination was of primary importance to teachers and stu-

dents alike. The teachers at Amber Hill did not make any

pretence of preparing students for more open, applied or realistic assessments of their knowledge. They were clear

that their job was to prepare students for the GCSE examination in the best way possible. The students were

also convinced of the aim of mathematics lessons and

they reported that the high degree of motivation and hard

work they demonstrated in lessons derived t’rom theil desire for examination success:

JB:

“So if you all dislike it so much. lvhy do you work so

hard in lessons’?”

C:

“Because we want to do well. maths GCSE is really

important. everyone knows that.” (Chris. Amber Hill. Year I I. Set 4).

The pressure the students received to do well at Amber Hill may have disadvantaged them in the examination in

the same way that the lack of pressure to do well may

have diminished the performance of the Phoenix Park students. However there were a number of indications

that the Phoenix Park students faced a range of real and

important disadvantages when they took their GCSE examinations. which the Amber Hill students did not

have to contend with. These factors made the results from the two schools somewhat surprising.

At the end of Year I I Amber Hill entered I82 of the

217 students in the Year group for GCSE mathematics.

this amounted to 84% of the cohort. At Phoenix Park IOX of the I I5 students in the Year group were entered for the examination, which was 94% of the cohort. The

two schools used different examination boards but Tables

6 and 7 give the results of the students at each school, as well as the national results for the different examination

boards. Table 8 shows the A C and A-G results from

both of the schools side-by-side. The GCSE results at the two schools show that similar

proportions of students at the two schools attained A*~~

C GCSE grades but significantly more Phoenix Park students attained A*-G grades. This was despite the simi- larity in the cohorts at the end of Year 8. the increased

motivation of the Amber Hill students, the ‘examination- oriented’ approach at Amber Hill and the lack of cal-

culators at Phoenix Park. Indeed, six Phoenix Park stu- dents wrote onto their actual examination papers ‘I haven’t got a calculator’ and at frequent points in the examination they wrote out the method they had used in the questions. but did not evaluate the answers. thereby losing marks.

During visits to the two examination boards I recorded

136 J. Boaler/Eualuation and Program Planning 21 (1998) 129-141

Table 6

Amber Hill GCSE results

Number of students

% of GCSE entrants

% of school cohort

National (%) average

A* A B C D E F G U X Y li

0 I 4 20 25 40 37 26 19 10 0 182 0 0.5 2.2 10.9 13.7 22.0 20.3 14.3 10.4 5.5 0 182 0 0.5 1.8 9.2 11.5 18.4 17.1 11.9 8.8 4.6 0 217 3.2 8.3 16.9 27.2 13.3 14.2 10.5 4.4 2.0 not available 100

Table 7

Phoenix Park GCSE results

A* A B C D E F G u X Y II

Number of students

% of GCSE entrants

% of school cohort

National (%) average

1 2 1 9 13 28 27 20 5 1 2 108 I 1.9 1 8.3 12 25.9 25 18.5 4.6 0.93 1.9 108 0.9 1.7 0.9 7.8 11.3 24.3 23.5 17.4 4.3 0.87 1.7 115 0.2 2.0 7.3 15.1 16.8 18.4 16.5 16.2 7.5 not available 100

Table 8

Comparison of GCSE results (%)

Entry

AH PP

Cohort

AH PP

A*-C 13.7 12.0 11.5 11.3 A*-G 84.1 93.5 70.5 87.8 % entered 84 94

the marks that each student attained for every question

on the GCSE examination papers. I had previously div- ided all of the questions into the categories ‘procedural’

and ‘conceptual’. Procedural questions were those ques- tions that could be answered by a simple rehearsal of a rule, method or formula. They were questions that did

not require a great deal of thought if the correct rule or method had been learned. An example of such a question would be ‘calculate the mean of a set of numbers’.

provided, of course, that students had learned how to calculate a mean, students did not have to decide upon a

method to use, nor did they have to adapt the method to fit the demands of the particular situation. An example of a conceptual question was ‘A shape is made up of four rectangles, it has an area of 220 cm’. Write, in terms of X, the area of one of the rectangles’ (diagram given). Such a question requires the use of some thought and rules or methods committed to memory in lessons would not be of great help in this type of question. All the examination papers, from both examination boards, included both procedural and conceptual questions in a ratio of 2 : 1, with twice as many procedural as conceptual questions. An analysis of the procedural and conceptual questions

that students answered correctly and incorrectly in each school reveals a significant difference between the

schools. The box and whisker plots given in Figs 1 and 2 show the distribution of the percentages of students

attaining correct answers for the two different types of

question at each school.

The conceptual questions were often, by their nature, more difficult than the procedural questions, even for a

student who had both learned and understood math- ematical rules and procedures. The students at both

schools would therefore be expected to answer more of the procedural questions correctly. At Amber Hill there

was a marked difference between the percentages of stu-

dents answering procedural and conceptual questions correctly, but at Phoenix Park the percentages of students

0

procedural conceptual

Fig. 1. Amber Hill.

25

0

procedural

j

conceptual

Fig 2. Phoenix Park

correctly answering the conceptual questions was. on

average, only slightly lower than the percentages solving the procedural questions. This analysis revealed some

important differences in the knowledge and under- standing of the students at the two schools. However, I was given special permission to conduct this analysis and such information is not normally allowed to be released,

to schools or anybody else. The significance of the differ-

ences revealed by this analysis and the other assessments I conducted. will be considered now.

3. Forms of knowledge

The different assessments reported in this paper were informative in showing patterns of behaviour that were

broadly consistent. The students at Amber Hill were able

to use the mathematical knowledge they had learned

when the requirements of questions were explicit. This meant that they could work through their exercises in

class with relative ease. they performed well on all of the

short, written tests that accompanied the applied activi-

ties I gave them and they were able to answer many of the procedural GCSE questions. The difficulties seemed

to occur for the students when the requirements of ques-

tions were not explicit, when they needed to use some mathematics after a period of time, when they had to apply mathematics and when they needed to combine different forms of mathematics. These ideas were both

illuminated and triangulated by two important forms

of evidence from the study-approximately 100 lesson observations in each school and in-depth interviews with approximately 80 students from the two schools.

In interviews the students at Amber Hill talked openly about their inability to use mathematics in anything other than clear, straight-forward textbook questions. When the students reflected upon their GCSE performance. they talked about the difficulties they faced applying

methods, remembering methods over time and com-

bining different areas of mathematics, for example:

L:

“Some bits I did recognise, but I didn’t understand how

to do them. I didn’t know how to apply the methods

properly.” (Lola, AH. Year 11. Set 3).

The difficulties the Amber Hill students experienced all

seemed to relate to. or fit within, an overarching phenom- enon that concerned the way in which students inter-

preted the demands of situations. In the architecture

problem a number of the Amber Hill students did not

calculate the volume or the angle of the roof correctly.

This was not because they could not perform calculations

with volume or angle. but because they needed to inter-

pret the questions in order to determine what to do. Many of the students were unsuccessful because they saw the

word angle and thought that they should use trig-

onometry. The Amber Hill students’ textbook model of mathematics teaching had not given them access to depth

of understanding so they were forced to look for cues in

mathematical situations that would tell them what they should do. When the students reflected upon the GCSE

examination they also related many of their difficulties to

the fact that the examination questions did not contain any cues in the way that their textbook questions did. In

the textbook questions the students always knew what

method to use-~-~the one they had just been taught on the

board. and if a question required something different or

additional to this. there \vas usually some clue in the question that would indicate what they had to do, In the

examination the students tried to look for similar clues but they could not find any:

G:

“it’s different and like the way it’s there like . not the

same.. it doesn’t like tell you it. the story, the ques-

tion, it’s not the same as in the books.. . the way the teacher works it out.” (Gary. Amber Hill. Year 1 1, Set 3).

The students experienced difficulty in the examination

because they found that the questions did not only require a precise and simplistic rehearsal of a rule, they required them to understand the questions and to know what the questions were asking them:

G:

“You can get a trigger, when she says like simultaneous

equations and graphs, graphically. when they say like. .and you know, it pushes that trigger. tells you what to do.”

JB:

” What happens in the exam when you haven’t got that?”

138 J. Boaler/E~aluation and Progrum Planning 21 (1998J 129-141

G: T:

“You panic.” (Gary, Amber Hill, Year 11, Set 3).

The students’ responses to the GCSE examination were

consistent with the mathematical behaviour they dem-

onstrated in a range of other assessments. This behaviour

had important implications for their use of mathematics in ‘real world’ and job situations. When the students

were in Years 10 and 11, I asked all of the students I interviewed to think of situations when they used math-

ematics outside school and to tell me whether they made

use of school learned methods in these situations. The

Amber Hill students, like the adults observed in other

research settings (Lave et al., 1984; Masingila, 1993;

Nunes et al., 1993) all said that they abandoned school

mathematics and used their own methods. But the stu- dents did not choose their own methods over their school-

learned methods only because they could not remember

or use school learned mathematics. They chose not to use school-learned methods because of the way they inter-

preted the demands of the ‘real world’. For the students at Amber Hill believed the mathematics they encountered

in school and the mathematics they met in the ‘real world’

to be completely and inherently different. When I asked

the students whether they believed the demands of the classroom and the ‘real world’ presented any similarities.

they all reported that they did not. The students analysed these differences in interesting ways, describing the math- ematics of the ‘real world’ as distinct because it was ‘social’, because it involved the self and because it had

meaning (Boaler, 1997a). The students’ views lend sup-

port to a situated perspective, because the students did

not use the mathematical knowledge that they could use in textbook situations precisely because of the way in

which the social setting of the ‘real world’ and their

relations within it, impacted upon them.

“I think it allows.. . when you first come to the school and you do your projects and it allows you to think

more for yourself then when you were in middle school and you worked from the board or from books.”

JB:

“And is that good for you do you think?”

T:

JB:

T:

The students were not inhibited in the way that the Amber

Hill students were. They were not struggling to remember specific procedures, nor search for cues which might indi-

cate the procedures to use. They were free to consider the

different questions and make sense of them:

JB:

A:

The students at Phoenix Park faced a number of dis-

advantages that may have diminished their examination

performance, but they still attained higher grades than the Amber Hill students. The reason for this appeared to

be that students could make use of the mathematics they had learned when it was assessed and even though they

had not covered everything they needed for the exam- ination, they could make effective use of the mathematics

they had encountered before. The superior performance of the Phoenix Park students on conceptual questions also provides an important clue as to the reason for their general success. The students were able to use math- ematics in different situations because of their attitudes towards and beliefs about mathematics. When the stu- dents approached questions, they believed that they should consider the situations presented and interpret what they needed to do:

The students at Phoenix Park had learned to interpret

mathematical situations, to make sense of the infor- mation they were given and to apply the mathematics

they knew:

JB:

G:

JB:

“Can you tell me anything you like about maths?”

The students at Phoenix Park remembered ideas and made use of them, they did not try and remember algo- rithms and procedures, nor expect to take methods as they had learned them and transplant them into new

“Yes.”

“In what way?”

“It helped with the exams where we had to.. . had to

think for ourselves there and work things out.” (Tina, Pheonix Park, Year 11, RT).

“Did you feel in your exam that there were things you hadn’t done before?”

“Well, sometimes I suppose they put it in a way which

throws you, but if there’s stuff I actually haven’t done

before I’ll try and make as much sense of it as I can, try and understand it and answer it as best as I can

and if it’s wrong, it’s wrong.” (Angus, Pheonix Park,

Year II, RT).

“How long do you think you can remember work after

you’ve done it?”

“Well I have an idea a long time after and I could probably go on from that, 1 wouldn’t remember exactly how I done it, but Id have an idea what to do.” (Gary, Pheonix Park, Year 11, MC)

J. Boah/Eraluation and Program Plamip ?I i 1998 I 12%141 I .?Y

situations (‘[ wouldn’t remember exactly how I done it’).

The students at the two schools had developed very different perceptions about mathematics that reflected

the differences in their schools’ ‘use and understand’, vs ‘transmission and transfer’ approaches. At Phoenix Park

the students seemed to have developed the ability to think holistically about the requirements of situations, as they

had been encouraged to do within their classwork projects. They were prepared to think about questions.

even if they did not know, or remember. any set pro-

cedures to use. The clear differences in the forms of knowledge and

understanding that the students developed at the two schools had important implications for their use of math- ematics in the ‘real world’. At Amber Hill the students

reported that they did not, or would not, make use of

their school-learned mathematical methods, because they

could not see any connections between the mathematics of the classroom and the mathematics they met in their

everyday lives. At Phoenix Park the students did not regard the mathematics they learned in school as inherently different from the mathematics of the ‘real

world’. When I asked the students at Phoenix Park about their use OE school-learned mathematics in the ‘real

world’, three-quarters of the 36 students said that they

regularly used the mathematics they learned at school because it was ‘the same’ as the mathematics of the ‘real world’.

Although the students at the two schools were only

giving their reports of their use of mathematics, these

reports were consistent with the mathematical behaviour

they demonstrated in other situations. The Amber Hill students’ descriptions indicated that they saw little use for the mathematics they learned in school in ‘real world’

mathematical situations and so they abandoned their school-learned mathematics and invented their own

methods. The students appeared to regard the ~t~o&is of the school mathematics classroom and the rest of their

lives as inherently different. This was not true for the Phoenix Park students who had not constructed bound-

aries (Siskin. 1994; Lave. 1996) around their school math-

ematical knowledge in quite the same way.

4. Discussion and conclusion

I have presented a range of forms of evidence within this article which has meant that the accounts of my data have been given in a highly summarised form. 1 will

attempt now to pull together the issues that have been raised by this somewhat disparate account, that I hope will justify the inclusion of the various forms of evidence given.

I became convinced during my three years of work in Amber Hill and Phoenix Park schools that the students

at the two schools had developed very different math-

ematical capabilities, perceptions and understandings.

These meant that the students at Amber Hill were less

capable in realistic assessment tasks, in which they needed

to use and apply the mathematics they knew; in situations when they needed to use mathematics some time after they had learned it; in examination questions that

required them to think about what they were doing and

importantly. in the ‘real world’. I have said that these

differences were indicative of the inert nature of the

Amber Hill students’ knowledge. a knowledge that they

found difficult using in anything other than textbook situations. At Phoenix Park the students spent less time

on task and they did not know LI.Y r?~uc,h mathematics, but

they had developed flexible forms of knowledge that they could use in a range of settings. I have related these

differences to theories of situated cognition, because the students at Amber Hill were unable to use mathematics

in the ‘real world’ because of the ways that they related

to aspects of this setting. The students described the ways that the social, relational. meaningful nature of the real

world stopped them from making use of the mathematics they knew. At Phoenix Park the students were successful

precisely because of their interpretations of experience and their willingness and capability to consider different

situations. reflect upon what they knew and adapt and

change mathematical methods to tit the demands of new situations. They did not transl’er set pieces of knowledge from one situation to another. they created knowledge ;I

new in every situation. by reflecting. thinking about and

adapting what they had learned.

The Amber Hill students were fazed by the math-

ematical demands of the ‘real world’. because they involved a social dimension, they held meaning and they related to them as people. These I’catures. the students

reported. stopped them from using their school learned methods. Hutchins has proposed that the “properties ot the interaction between individual minds and artifacts of

the world” are at the essence of human performance ( Hutchins. 1993). Traditional methods of assessment do

not assess this interaction. Nor do they assess a students’ ability or predisposition to form meaning in ~IJ/O/;O/I JO

new situations (Lave. 1996). evidence from this study

suggests that this meaning devclopmcnt is central to mathematical competence.

The responses and mathematical behaviours of the

students in the two schools raise some obvious and important implications for teaching and learning styles (Boaler. lYY7a). but they also raise questions about the information that is provided by dominant forms of

assessment in schools. For example. the only important assessment of the mathematical knowledge and under- standing that students develop in schools in the U.K., is the GCSE examination. This national examination. taken when the students are aged 16 years, is used as an entry requirement for higher courses of study as well as an indication ofjob performance for students who choose

140 J. BoalerlEwluation and Program Pluming 21 (1998) 129-141

to leave school and seek employment at this time. The

important and clear differences in the capabilities of the Amber Hill and Phoenix Park students that were shown

by their performance on applied tasks, long term tests and conceptual questions, their reflections in interviews

and their behaviour in class, were largely hidden within

the results of the GCSE examination. Consider, for exam-

ple, the highest attaining students in each school. The proportion of the Year groups at Amber Hill and Phoenix

Park that attained the top grades in the GCSE exam- ination, grades A*-C, was exactly the same, 1 I%, but

the Amber Hill students attained their grades by solving

procedural and conceptual questions in the ratio of 2 : 1, whereas the Phoenix Park students solved equal pro-

portions of each type of question. This difference was

extremely important, because it showed that the Phoenix Park students were more capable at solving non-standard

problems and interpreting mathematical situations. This

competence could be taken as a better indication of the students’ capability in ‘real world’ situations than the

ability to reproduce standard algorithms in standard examination questions. Students’ reports of their ability

or predisposition to use mathematics in ‘real world’ situ- ations add support to this claim.

The different sources of data that formed part of this

study showed, quite clearly, that the students at the two

schools developed forms of mathematical knowledge and

understanding that were fundamentally different. The results of the GCSE examination failed to reveal this

difference. My analysis of the types of questions that students answered correctly within the GCSE exam- ination was informative, but I was given special per-

mission to see the students’ examination answers and this sort of information is not released to schools. Messick

(1988) has talked about the meanings and consequences of assessment and Wiliam (1996) has described the ways that forms of assessment ‘canonise certain aspects of

knowledge’ (p. 132) and define ‘what is important, what is worth knowing’ (p. 132). This, in my view, is the fun-

damental problem associated with traditional assess- ments such as GCSE examinations. It is not that they do not differentiate for certain forms of knowledge, but that they allow and therefore encourage forms of knowledge that are shallow, procedural and ultimately limited. Tra-

ditional models of teaching have been proved effective at training students for traditional models of assessment but such teaching methods encourage forms of knowledge that are shallow and transient and easily exposed when subject to more realistic and meaningful assessments or the demands of the workplace.

Phoenix Park provides a rather salutary example of the consequences of the GCSE mathematics examination that is used in England. Five years ago Phoenix Park was involved in a pilot of a new GCSE examination that combined open and closed questions, in order to assess mathematical process as well as content. Students were

invited, in this examination, to investigate an extended

mathematical problem and solve a number of conceptual, applied questions. In 1994 the School Curriculum and

Assessment Authority withdrew this new form of GCSE examination, because they wanted to impose single mod-

els of examination across the country. The next cohort

of Phoenix Park students was required to take the more traditional examination. The proportion of students

attaining grades A-C and A-G shifted from 32% and

97% respectively in 1993 to 12% and 84% in 1994. Shor- tly after this, at the end of my research project, the head- teacher responded to the fear of forthcoming government

inspections and the need to improve examination per-

formance by forcing the mathematics department to abandon their project-based approach and teach from

textbooks. It seemed that a project based approach could be justified when it was assessed through an open exam-

ination, but it was difficult for the teachers to support this approach (even with my research evidence) in the midst of a back-to-basics educational climate and a tra-

ditional, closed GCSE examination. The consequences

of this traditional model of GCSE assessment were to force one of the only remaining schools in the U.K. that used an open, thinking-based mathematical approach to

return to textbook teaching.

The results from this study and a number of other recent research studies (CTGV. 1990; Brown et al., 1989)

show that the perspective of situated cognition has enor- mous implications for educational progress. Yet policies

that impact upon schools in the U.K. are encouraging subject departments to move further towards formalised,

textbook models of teaching and away from approaches that may acknowledge, if only slightly, the situated nature

of human cognition. The role of assessment is crucial to this development. The most informative assessments that

I conducted at the two schools were the applied tasks, that I followed with interviews and my analysis of the

conceptual and procedural GCSE questions, that I was

given special permission to conduct. Neither of these assessments, that provided essential information about the students’ use of mathematics and capability to inter-

pret situations and develop meaning from them. are options that may be used to monitor and report on school students’ understanding. Yet the capabilities that were assessed within applied tasks and conceptual questions

are widely acknowledged as being central to the devel- opment of ‘real world’ capabilities, amongst researchers (French, 1992; Noss, 1994) as well as industrialists (Pea- cock, quoted in Ball, 1990).

The restrictive policies in teaching and assessing that are now in place in the U.K. may have serious impli- cations for the workforce of the future. The consequences of this shift in assessment at Phoenix Park were to end an approach that promised to acknowledge, at least partly, the situated nature of human cognition. They also served to eradicate an approach that gave students a

genuine mathematical empowerment, that they could use,

both within and outside their mathematics classrooms.

Acknowledgements

I would like to thank the ESRC who funded this research study. Paul Black and Mike Askew who gave

me help and encouragement throughout the study and

Dylan Wiliam for his advice on w/riting this article.

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