Mathematics Education Research Journal
1999, Vol. 11, No.2, 109-123
Teaching Mathematics: A Brightly Wrapped
but Empty Gift Box
Sandy Schuck
University of Technology, Sydney
In a research study conducted with first year prospective primary school teachers,
beliefs about and attitudes towards mathematics were investigated. It was found
thatprospective teachers generally perceived good mathematics teaching to include
the provision of enjoyable experiences. However, most of the student teachers did
not perceive enjoyment as having an intellectual component. Further, they often
expressed the belief that being knowledgeable about mathematics was a
disadvantage for a primary school teacher. It was as if student teachers see
mathematics teaching as brightly coloured wrapping around an empty box.
Implications of these findings for mathematics teaching and teacher education are
discussed.
This paper is the fifth in a series of papers regarding prospective teachers'
beliefs about the learning and teaching of mathematics (Schuck, 1996a; Schuck,
1996b; Schuck, 1997; Schuck, 1998). The papers are based on a study, conducted at
an Australian university, which investigated the following research questions:
•
•
•
What beliefs and attitudes about mathematics and mathematics education
do first year prospective primary school teachers bring to their tertiary
education?
How do these beliefs and attitudes affect the learning of mathematics and
the learning about mathematics education in teacher education courses?
How do students' beliefs and attitudes affect their ideas on good practice
in the teaching of mathematics in the primary school?
The present paper focuses on two aspects of the study: teachers' views of (a)
the importance of subject matter knowledge, and (b) the contribution of "fun"
activities to effective teaching.
Subject Matter Knowledge
Subject matter knowledge has been the topic of debate amongst educators for
many years. Schmidt and Kennedy (1990, p. 1) suggest that educational observers
have tended to portray the reforms in education in terms of a "fundamental
tension between teaching specific facts, on one side, and fostering independent
thought on the other"-a tension between content and process. This view of the
reform process could be constructed to mean that subject matter knowledge is in
opposition to independent thought, rather than being a necessary part of such
thought. Constructivism, and other theories of learning which hold that
knowledge is not gained in a passive manner from a teacher-expert but rather is
constructed actively by learners as a result of their experiences, have been
interpreted by some educators as implying that there is no need for the teacher to
110
Schuck
be an expert in the subject matter. These educators see their roles as providing
rich environments for students and acting as facilitators of learning (Mosenthal &
Ball, 1992). Consequently, in some teacher education programs, the desire ·to
challenge students' beliefs about learning and teaching, and to shift these beliefs
away from transmissive models and towards constructed models, may be
responsible for a downgrading of the value of subject matter knowledge in these
programs (Hoden, McDiarmid, & Wiemers, 1990).
However, many educators see subject matter knowledge as essential for
effective teaching. While agreeing that the affective component of learning is as
important as the cognitive component, and that there is a need to challenge beliefs
that teaching is telling and providing drill and practice, they suggest that it is vital
for teachers to have a deep understanding of disciplinary subject matter. They
agree that helping school students develop a rich understanding of subject matter
cannot simply be done by telling them information and having them repeat it.
Teachers must be able to view the subject matter through the learners'
perspectives, and guide the learner with the aid of their own understanding of the
content (Dewey, 1964; Feiman-Nemser & Buchmann, 1986). Sullivan and Mousley
(1996, p. 76) put it this way: "BUilding understanding suggests the development of
concepts and mathematical ideas as well as developing conceptual links. ... It
requires an orientation on the part of the teacher to plan and teach mathematics in
an orderly, coherent, and connected way which supports sound, flexible
conceptual development and elaboration of mathematical structures./I
Shulman (1986) developed the concept of content knowledge, defining a
number of different knowledge categories which are essential for teachers:
subject matter knowledge, including understanding the underlying
structures and concepts of the subject;
• curricular knowledge, including knowledge of what comprises the
curriculum and what materials are associated with curricular knowledge
as well as other knowledge about the curriculum; and
• pedagogical content knowledge, including knowledge of the content that is
most appropriate for children to learn, knowledge of which avenues are
fruitful to explore, and knowledge of what aspects of the subject matter
tend to be problematic.
•
This paper is arguing that all three forms of content knowledge are essential for
teachers, and that without subject matter knowledge it is extremely difficult to be
knowledgable in the other two areas.
Further support for this argument is given by the authors of the North
American publication, Professional Standards for Teaching Mathematics (National
Council of Teachers of Mathematics [NCTM], 1991), who clearly believe that
subject matter knowledge is essential for teachers. They suggest that teachers
should have a range of knowledge about mathematical concepts and procedures
and that teachers need this knowledge in order to decide how best to help their
students learn mathematics.
In initial teacher education, the value of subject matter knowledge has also
been emphasised. The US' National Commission of Teaching and America's
Teaching Mathematics: A Brightly Wrapped but Empty Gift Box
111
Future (Darling-Hammond, 1998) concluded that the reform of elementary and
secondary education depends on increasing teachers' access to knowledge to meet
the demands they encounter. Darling-Hammond discusses the large body of
research that confirms that knowledge of subject matter is an important element
ofteacher effectiveness. She asserts that teacher education courses are extremely
iniportant in preparing teachers both in discipline knowledge and in knowledge
about teaching. The Australian report, The Discipline Review of Teacher Education in
Mathematics and Science (Department of Employment, Education and Training
[PEET], 1989), claims that many students entering initial primary programs have
only a superficial knowledge of mathematics content. The report emphasises the
irn.portance of student teachers being mathematically competent and having
higher order mathematical knowledge to give them confidence in their teaching
and to provide them with tools with which they will be able to help their future
students. Taplin (1998) suggests that these recommendations direct us to examine
our initial teacher education programs to ensure that student teachers are
developing the necessary skills, concepts, and procedures that will allow them to
be effective teachers. Borko, Eisenhart, Brown, Underhill, Jones, and Agard (1992)
describe the tension experienced by a novice teacher when pushed to explain to
students why a particular procedure worked. The authors conclude that the
teacher's lack of conceptual knowledge and lack of desire to engage in difficult
mathematical thought destroyed the opportunity for the students to experience
rich and deep understanding of a topic.
The debate is not just a simplistic argument about the merits of teaching
specific facts versus teaching to develop autonomous thought. Kennedy (1990)
suggests that people who are knowledgeable about a subject know more than just
facts and ideas; they have also formed the connections between these ideas and
further understand how to approach new problems and produce new ideas
within the subject. They are knowledgeable about the content, the organisation·
and structure of content, and the methods of inquiry in that subject. In this way,
having strong subject matter knowledge is part of being an independent thinker.
However, Mayers (1994) and Foss and Kleinsasser (1995) state that when
preservice teachers mention improving their own knowledge of mathematics, the
knowledge to which they generally refer encompasses arithmetic computation
and little beyond the basic skills needed in the primary schooL The view of
knowledge held by student teachers is important because it will influence the way
the students eventually teach. Ball and McDiarmid (1990) state that teachers'
conceptions of knowledge shape the questions, tasks, and activities they develop.
The authors went on to say that teachers need to know more than the specific
topics of their curriculum. Teachers need to be able to help their students define
the means of inquiry in that field, and to see the connections to other areas. They
need to have a range of strategies for developing understanding of the topic.
Sullivan (1992), while agreeing that learning which emphasises connections and
recognises students' experiences is likely to be stronger than learning based on
rules or taught procedures, suggests that the implications for the teacher of these
approaches are complex. He points out that we cannot assume that students will
gain the full range of desirable experiences simply by conducting undirected
Schuck
112
explorations. Some teacher guidance of the learning is necessary.
Teaching for Enjoyment
One of the goals for mathematics education listed in the National Statement on
Mathematics for Australian Schools (Australian Education Council [AECl, 1991) is
that students should develop confidence and positive attitudes towards
mathematics. The authors see it as important that students should gain pleasure
from doing mathematics and seeing its relevance.
These goals are shared by other policy documents on the teaching and
learning of mathematics (DEET, 1989; NCTM, 1991) which state that students
should enjoy the mathematics they experience in schools. To this end, they
recommend that teaching should be motivating, relevant, participatory, and
accessible to all..
However, in a review of the literature on affective factors in teaching primary
school mathematics, Carroll (1998) found that many studies reported that both
inservice and preservice primary school teachers were generally negative in their
attitudes to school mathematics. The question raised in some of these studies was
whether teachers holding such attitudes were capable of teaching mathematics for
enjoyment.
Ironically, both Ball (1988) and Foss and Kleinsasser (1996) found that
American preservice elementary teachers believed that "making mathematics fun"
was central to the learning process. However, these authors also found that
students expected that enjoyable lessons would succeed in teaching mathematics
regardless of the content or context of the lessons. It appeared that teachers were
well aware of the need to make mathematics lessons enjoyable, but believed that
the most effective way to do so was to downplay the mathematical content of the
lessons.
Little research has been done in Australia on the topic of teaching
mathematics for enjoyment in primary schools. My research on this topic
indicates that student teachers often strongly support the notion that the teaching
of mathematics should be motivating and enjoyable, but are often unsure of how
this motivation may be aroused (Schuck, 1996b).
Method
Participants
Fifty students participated in the year-long study; all were in their first year in
an initial primary teacher education program at an Australian university. The
participants were all members of two tutorial groups. These two groups were
selected for the study from the four first year tutorial groups because I was their
tutor in their first semester and was, therefore, able to build up a positive
relationship with these students.
Of the participating students, approximately 65% were recent schoolleavers and
the remaining 35% were mature-age entrants (i.e., they had been out of school for more
than three years). Over 80% of both the mature-age students and the recent school
Teaching Mathematics: A Brightly Wrapped but Empty Gift Box
113
leavers were female. Over 80% of the participants had studied mathematics for all
13years of their school education.
Four mathematics teacher educators (three males and one female) who had
been involved in the development and teaching of the mathematics education
course also participated in the study. All had been teacher educators for over ten
years.
Data Collection
The research was an interpretive case study. A grounded theorising paradigm
(Strauss & Corbin, 1998) was used, in which my theorising about students' beliefs
developed from the data and influenced the direction of future data collection. There
were four phases to the research:
•
•
•
•
Phase 1. Data were collected at the beginning of the first semester of the
students' first year through interviews formulated and conducted by the
students themselves. Each students also wrote a reflective critique of the
responses to their interview questions.
Phase 2. Data were collected by means of an open-ended questionnaire
given to students towards the end of their second semester.
Phase 3. In-depth interviews with eight students comprised the Phase 3
data. These interviews took place after the close of the second semester of
the students' first year of the teacher education course or at the beginning
of the following year.
Phase 4. In-depth interviews were conducted with the four teacher
educators responsible for the mathematics education subjects.
The data for Phase 1 came from activities designed to enable the students to
gain insights into their own and their peers' beliefs (Schuck, 1997). To prevent my
interests from dominating the data, I requested the students to develop interview
questions about issues that were important to them. They then asked a partner
these questions, and in return were asked questions by their partner. The students
were then required to reflect on the implications of their partner's responses for
the learning and teaching mathematics. The data comprised the questions, the
responses to the questions, and the reflective critiques.
In order to meet the stringent ethical criteria developed by my university, I
delayed analysing the first semester data until I was no longer the students'
lecturer. In the second semester, I ensured that I had no contact with these
students other than as researcher.
The themes and issues that recurred in the students' interviews in Phase 1
became topics of questions which probed these issues further in Phase 2. For
example, in Phase 1 students had repeatedly mentioned "having fun" as an
important part of teaching. In the questionnaire, this issue was investigated in the
question "Many students stated in the interviews that they believed mathematics
should be fun. What is your view? Please explain first what you understand by
the expression 'mathematics should be fun'," The questionnaire had three parts to
it: The first part asked for demographic data from the students, the second part
114
Schuck
investigated the issues that had arisen in Phase I, and the third part explored
students' experiences in the mathematics edu<;ation subjects they had studied in
their first year. All questions, apart from the demographic details, were openended. The entire questionnaire appears in Schuck (1996a).
The Phase 3 interviews were developed from the data already collected, to
serve as a method of triangulation and also to provide me with the opportunity to
develop any response further. (This was actually the first occasion on which I
interacted on a one-to-one basis with participants.) It had become of interest to me
to see if attitudes and beliefs of participants were affected by their age or by the
amount of reflection they appeared to do. Consequently, the eight students
participating in Phase 3 were selected by purposeful sampling (Bogdan & Biklen,
1992). Firstly, students were chosen so that there would be equal numbers of
mature age students and recent school leavers. The mature age students had left
school over a period ranging from 4 to 35 years previously. Secondly, students
were chosen so that there would equal numbers of students who appeared to be
reflective thinkers and those who did not show evidence of such thinking.
Students who gave thoughtful and deep responses to questions in the first two
phases were regarded as having demonstrated reflective thinking; those who gave
brief and superficial answers were regarded as not having demonstrated
reflective thinking. Two mature age students and two recent school leavers were
chosen from each reflectiv~thinkingcategory. The full interview schedule appears
in Schuck (1996a).
Finally, the four mathematics educators were interviewed to investigate their
rationale for choosing the content of the first year mathematics education subjects.
Their philosophies on the epistemology of mathematics and their views on the
role of the mathematics education subjects were also explored. The interview
schedule for Phase 4 also appears in Schuck (1996a).
Data Analysis
The data were analysed using common procedures for analysing qualitative
data (Miles & Huberman, 1994). The first step involved coding the data and making
notes on the data. The next step was to search for similarities in phrases and
themes, relationships between variables, and differences or extremes in the data.
Finally, the themes were examined as to whether they suggested any
generalisations that could be made, and these generalisations were tested against
the consensus of accepted knowledge.
The data were first coded in a general way directly after each phase of data
collection. After all the data had been collected, the data were coded again in a
more precise way-drawing out common themes, noting alternative views, and
forming relationships between variables. There was strong agreement between the
first and second sets of data coding, providing triangulation of the analysis. In the
second coding, constant comparison was used to link themes with those already
present; new categories were formed as necessary. Themes and commonalties
were examined further and a theoretical framework developed. The analysis was
done with the aid of the computer software tool NUD-IST (Quality Solutions and
Teaching Mathematics: A Brightly Wrapped but Empty Gift Box
L
l
115
Research, 1997), which allowed me to readily store, retrieve, and code large
amounts of data.
Results
Making Sure Mathematics Lessons are Fun
In Phase 1, many students talked (without prompting) about the need to make
mathematics fun:
Ithink children learn maths best by having it made enjoyable to them. If maths is
presented through activities, games and other group and individual workshops,
children are likely to develop an enthusiasm for maths and hopefully then
curiosity will increase.
The best way to teach and for children to learn maths is to create an enjoyable
atmosphere so as to make maths "fun".
If you teach students in a way that creates games and activities they seem to
remember it more. They enjoy it a lot more which improves their attitudes
towards maths and helps them learn more. Also just to make the lesson more
exciting for them, it could help for them to learn that way too.
Bridget explained how her past experiences had led to this emphasis on having
fun in mathematics:
Our conception of the way in which we were taught maths seems to be that it was
hard, dull and boring and that this needs to be rectified in order to make maths as
interesting and as much fun as possible. Thus ensuring that our future students
gain a better understanding of the basics of maths and a positive attitude to the
learning of it.
Different students had different conceptions of what "fun" was: To some it
was the playing of games-in contrast with their _school experiences of sitting in
silence, completing textbook exercises; others had more profound meanings for
fun. For example, Christine, a recent school leaver, wrote:
Maths needs to be fun, and made able to relate to. I believe if maths is taught in a
fun, practical sense, by using maths-related games and group work it would be
far more
beneficial in the long run.
.
.
By contrast, Gail, a mature age student who was at school over thirty years ago,
explained what fun meant to her as follows:
•
•
•
•
•
That there should be involvement and participation.
Being fun does not mean to me a constant flow of tricks and puzzles.
A child (AND A UNI STUDENT) can only have "fun" when slhe is
understanding and achieving-there is no fun in being overwhelmed and
failing.
Fun implies to me a teacher who realises that a concept must be presented in a
variety of ways in order that people of all levels grasp that concept.
Fun means - never being humiliated for being wrong or not knowing how to
do a question!
116
Schuck
Gail later elaborated her views further:
I
Well, I think it's more than important, I think it's actually essential and fun,
meaning, I said this in my questionnaire, that fun to me isn't just frivolity and
laughter, fun is that freedom to understand, to conceive something, to grasp it.
That to me is fun, I love that.
So to Gail, fun occurs when the classroom environment is supportive and
encouraging of learning and has an intellectual component. The role of the teacher
also appears to be more than merely a facilitator of enjoyable activities.
Agnes saw fun as being the antidote to her experiences in the mathematics
classroom:
What I believe the statement suggests is that mathematics should be able to be
experienced and enjoyed. And that it doesn't have to live up to its expectations
[in] society of being boring and completely "black and white". Children should
be able to explore and discover with mathematics - something I did not get an
opportunity to do.
Lexi viewed fun in a similar way:
Mathematics should be fun and enjoyable. I believe the student teachers were
referring to mathematics not being difficult, useless and boring but enjoyable,
challenging and interesting.... Children who enjoy mathematics and think it is
"fun" will continue to attempt mathematics in future years.
It appeared that fun was required to prevent mathematics from being boring;
most students seemed to have found mathematics to be very dry and boring in
their school days. Feelings similar to the following were expressed repeatedly:
Saying that "mathematics should be fun" indicates that a child should be keen to
learn and not try to avoid it because it is boring. Mathematics should be taught in
a variety of ways and should involve using concrete materials wherever possible.
Fun activities should be incorporated into the lessons to keep the children
interested.
This theme of having fun was very strong. Most of the students hoped to
make lessons more interesting by introducing games, relevance, or group work.
Concrete materials and exploration were also suggested. Only Gail expressed the
belief that understanding mathematical concepts could itself be fun.
Teachers' Knowledge ofMathematics
Many students talked about the role of mathematical subject matter
knowledge. Pip spoke about the advantages of having little background
knowledge in the following terms:
I won't come in with any extra baggage that I have picked up all the way through
high school and this is ... why I think I click with little kids, with the kids, because
I'm still on their level, with a lot of things .... We [mature age students] don't
have the HSC [final year of school] knowledge to fall back on..,. [But] it didn't
bother me because I thought I could handle [it]., I know I can do up to the level of
maths that primary can do.
I know just from personal experience I don't ... because of my lack of
Teaching Mathematics: A Brightly Wrapped but Empty Gift Box
117
knowledge I don't get agro [aggressive/angry] if they don't get it right. I can sit
down for hours and help them work a sum out and show the different ways of
working it out without losing my temper, because I don't have this ... baggage of
e"tra knowledge on me. Because I didn't have this knowledge, is why I was told I
was stupid. "You should know how to do it, you're an idiot." But I didn't have
the knowledge to do it ... so you know, I think it's an extra plus that I've got, to
take into a classroom.
Pip thus saw her lack of knowledge as an advantage to her teaching, because it
would make her more patient and sympathetic if her students struggle to
understand. Rene shared Pip's viewpoint:
I feel that my lack of understanding in particular areas at primary school will aid
my teaching of that area. For example I will have a better knowledge of teaching
fractions and of its practical applications. This will enable me to understand and
react appropriately to the problem the students may have.
Aaron related how having a teacher who was a brilliant mathematician had
caused him to battle with mathematics at school:
He was a genius at maths, like he could do anything. Especially when you're in
year 8, you see this guy who can do these amazing things with mathematics and
you think "God, who is this guy?" ... But he just couldn't ... I suppose he was
frustrated with us dills who just sat there but ... he was brilliant at maths, he just
couldn't get it across, unless you were brilliant as well, then he got through to
you with his method but uh [shakes his head] .
Sally expressed a similar opinion:
I think that background knowledge is important but the knowledge of teaching
procedure is more important. Many people are good at mathematics but because
it has always come easily to them, are not so good at explaining it to others.
Several students appeared to believe that their difficulties with a teacher arose
because of the teacher's ability in mathematics rather than because of any other
characteristics of the teacher.
On a similar theme, the questions some students asked in the paired
interviews in Phase 1 indicated they had doubts about the value of being
confident about mathematics. For example, Dale asked John:
At primary school you were confident ... will that make it easier for you to teach
and empathise with students that have less ability than you or won't [it?]
This theme was also picked up by other students whose partners seemed to have
high mathematical ability. Terry wrote of his partner Clive:
Clive's attitude towards mathematics reflects the way he will teach mathematics
in the future. Maths is an important subject to him and therefore he will teach it
enthusiastically which often influences the students' attitudes towards the subject.
Although Clive did well in this subject he realises the importance of developing
these skills and will use a variety of methods to ensure the students'
understanding.
It can be seen that Terry believed that attitude was extremely important in
118
Schuck
teaching and that being enthusiastic was vital for a good teacher. In this respect,
she saw Clive's ability as being beneficial because it would promote enthusiastic
teaching. However, she also seemed to feel his ability had drawbacks, as she
prefaced her next sentence with a reservation ("Although Clive .. .").
Dale also expressed reservation about John's mathematical ability and its
implications for his teaching:
His experiences in this subject at junior school were generally very positive,
apparently as a result of his competence as a mathematician.... John considers
some of the new innovations in maths teaching useful, but overall he seems to
prefer more "classic" lessons, centred around individual students using
workbooks or similar resources. He feels that the brighter students will
automatically enjoy maths, but will use some new techniques to interest and
involve those of lesser ability. John is well aware of the need to empathise with
struggling students, although occasionally he might find this difficult. (Emphasis
added)
This opinion-that being good at mathematics was not necessarily helpful for
teaching mathematics in primary school-was expressed by students who were
good at mathematics as well as by those who were not. Cara expressed some
awareness of the need to be sensitive to others' needs, even if they differed from
her own:
Because I picked it up so easily, I think that will make it hard for me to teach it in
a way, because I have trouble understanding how hard it is for some kids to pick
it up. So I'll really have to concentrate on giving everyone a fair go, trying to
explain as best as I can so really it's giving them a fair opportunity to understand
the work in their own time.
Those who suffered from a lack of confidence in their own ability in mathematics,
comforted themselves that it was not all that important to be able to do
mathematics:
I'll think "Oh God, I've got to teach maths now" and get all uptight about it. I've
got to say it's not hard, instead of thinking it's gonna be hard. It is, it's attitude
and so I'll have to ... It doesn't matter if you're not good at it. (Emphasis added)
. There were a few students who felt that their knowledge and ability were
advantages:
I guess my ability to understand a lot of the maths throughout school enabled me
to enjoy it more easily, whereas people who struggle with maths don't often see
the advantages of knowing mathematics.
Another student indicated a belief that lack of ability in mathematics was a
disadvantage for teaching:
As a result of these experiences, April expresses her reluctance to teach maths in a
primary school. It could be said that this may be due to a lack of confidence in her
own mathematical aptitude.
In general, however, most of the students did not appear to value knowledge of
mathematics-either as a purpose for mathematics education courses at
Teaching Mathematics: A Brightly Wrapped but Empty Gift Box
,
119
university or as a characteristic of a good teacher. In fact, they seemed to feel that
Stlchkrlowledge was a disadvantage to teaching in the primary school, as it would
pe>thprevent them from empathising with students who were struggling and
ihterferewith their ability to explain clearly.
Discussion
Teaching for Enjoyment
The findings of this study correspond with those of Ball (1988) and Foss and
Kleinsasser (1996). The finding that student teachers place considerable
importance on the pedagogical principle of making mathematics lessons enjoyable
thus appears to be generalisable across two different countries.
The present study clarifies what student teachers might mean by "having fun"
in a mathematics classroom context. The participants spoke about teaching in a
way that encourages the school students' active participation in learningthrough, for example, use of varied practical activities, interesting and easy
challenges, games and puzzles, and group work. All these techniques were
believed to lead to enjoyable lessons.
In a sense, the students' opinions were fully in line with reformist
recommendations (AEC, 1991; DEET, 1989; NCTM, 1991). However, the data did
not seem to indicate that students saw any link between the provision of enjoyable
classroom activities and the extraction of mathematical principles from those
activities. The practical pedagogy suggested by the students appeared to have the
sole goal of making the learning process fun, rather than promoting learning; and
it was the experience of having fun that was seen as leading to learning, rather
than the content of the practical activities themselves.
Subject Matter Knowledge
A major finding of the present study is that many of the participants did not
believe that good teaching was dependent on being knowledgeable about
mathematics; on the contrary, these student teachers believed that a strong
knowledge of subject matter might cause them to lose their empathy with
struggling school students. Many students had experienced situations at school in
which they had been taught by teachers who appeared to be talented
mathematicians but lacked effective teaching skills. These experiences had led to a
perception by students that having a deep subject matter knowledge would lead
to an inability to teach effectively; they seemed to feel that teachers who were
talented at mathematics would not have the patience, nor the communication
skills, to explain concepts to students who were struggling to understand.
However, research by Shulman (1986), Ball (1992) and others has shown that
it is necessary for teachers to have a strong subject matter knowledge in order to
be confident, and enthusiastic, and to be able to guide children down fruitful
paths of exploration and to know when to suggest that alternative routes are
taken. The view that primary teachers do not need to have a thorough
understanding of the mathematical priIi.ciples underlying school mathematics is in
Schuck
120
direct conflict with such research.
The notion that not understanding a topiC; will enable a teacher to better
understand the way that children will learn the topic is a disturbing one. The
attitude is analogous to that of a person who, seeing that someone has fallen
down a deep well, decides to show his or her sympathy by also falling down into.
the well. Surely it would be better to fetch help to get the person out of the welL
While it may be true that a teacher is more sympathetic and understanding of
children's difficulties if he or she has experienced similar difficulties, more than
empathy is required to help students understand better. A thorough knowledge of
the principles underlying the mathematical content with which they are
struggling, of how children learn, and of alternative strategies for developing
understanding is necessary if a teacher is to guide children out of their difficulties.
Conclusions and Implications
It appears to me that student teachers' beliefs about the virtues of fun and
lack of knowledge stem from the same source: feelings about their own
mathematical ability. The data show that students' past histories had often left
them anxious and fearful about their mathematical ability. The belief that being
knowledgeable about mathematics was a disadvantage for teaching helped to
justify their weakness in the subject and assured them that it would not prevent
them from being the effective teachers they so strongly desired to be. To resolve
the dilemma of wishing to give their own students a different experience of
mathematics from the one they had undergone, while at the same time disliking
and fearing the subject matter, student teachers intended to devote time and effort
to filling their classrooms with fun activities.
To consider good teaching to be purely about having fun, providing practical
activities and a supportive and enthusiastic classroom atmosphere with little
attention to the learning outcomes, is surely a hollow vision. Such teaching may
be seen as offering children a brightly wrapped but empty gift box. While the
offering appears to be an enticing and attractive gift, when it is unwrapped and
examined further it entirely lacks content. The deep understanding of
mathematical concepts is missing from the parcel, and it is this gift that could be
most beneficial to the student.
The implications of treating fun above knowledge are many. Firstly, if
student teachers do not view knowledge of mathematics as a desirable and
necessary component of their studies, little improvement in their mathematical
knowledge will occur. Borko et aL (1992) show this clearly in their case study of a
teacher wh9 desired to be an exemplary mathematics teacher, but was not
prepared to engage in the cognitive efforts needed to create understanding. If
student teachers value only the caring, nurturing side of teaching, they will not be
motivated to improve their own subject matter knowledge and the
recommendations of the Discipline Review (DEET, 1989) will not be achieved.
Secondly, if teachers develop activities which are entertaining and enjoyable
but do not see the need to extract the mathematical principles underlying these
activities, they will miss many opportunities to help their pupils gain in
Teaching Mathematics: A Brightly Wrapped but Empty Gift Box
121
understanding. If the implicit mathematical content is not made explicit, lessons
rnigl1twell be active and participatory but little mathematics will be learned. The
I~~ultwill be exactly what student teachers wish to avoid-pupils becoming
'icorl.ft.lsed and bewildered when mathematics teachers later assume they have
mastered the basic concepts.
1birdly, those student teachers who feel marginalised in terms of their
mathematical learning and power are unlikely to gain access to the power and
beauty of mathematics jf they devalue the mathematical knowledge to which they
are exposed. A vicious circle is then created in which their future students are also
unlikely to access the majesty of mathematics.
What do our results mean for teacher education? It appears essential that
teacher educators be aware that their students might well believe that empathy
alone, rather than subject matter knowledge, will make them good teachers. They
can then take steps to make students' views on mathematical knowledge explicit
and subject to challenge. Student teachers should be given opportunities to see the
value of the mathematical knowledge underpinning primary school mathematics,
and to become aware of the distinction between showing empathy for students
who are struggling and not understanding the work themselves. It must be
shown, for example, that having little understanding of fractions is detrimental
and not beneficial to one's teaching of the topic. Case studies such as the one
presented by Borko et al. (1992) might be useful discussion material here.
Role models who are both good at mathematics and also demonstrate
exemplary teaching could be used to counteract the belief that a teacher cannot be
good both at mathematics and teaching mathematics. The stereotype of the
brilliant mathematician who cannot teach should be analysed for its ability to lead
student teachers to the wrong conclusions. Surely it is not the brilliance that
should be discarded but the ineffective aspects of teaching practice. A discussion
of students' past experiences when supposedly brilliant mathematicians taught
them poorly can be used to analyse those teaching characteristics which exemplify
poor teaching. By postulating the opposite behaviour, the student teachers can
then develop their ideas of more effective teaching.
Finally, the pedagogy of participatory, relevant and enjoyable activities that
many student teachers espouse should be encouraged. At the same time, many
opportunities for drawing out and' discussing the mathematical content of the
proposed activities should be given. By analysing various units of work suggested
in the school curricula both for participatory value and mathematical content,
student teachers can be helped to see that it is possible to achieve both aspects in
their lessons. In this way, the brightly wrapped gift box can become filled with
content of substance.
References
Australian Education Council (1991). A National Statement On Mathematics for Australian
Schools. Melbourne: Curriculum Corporation.
Ball, D. L. (1988). Knowledge and reasoning in mathematical pedagogy: Examining what
prospective teachers bring to teacher education (Doctoral dissertation, Michigan State
University, 1988), Dissertation Abstracts International, A50(02), 416. (University
122
I
Schuck
Microfilms No. AAT8900008)
Ball, D. L. (1992). Magical hopes: Manipulatives and the reform of math education. The
American Educator, 16(2), 14-18,46-47.
Ball, D. L., & McDiarmid, G. W. (1990). The subject-matter preparation of teachers. In
W. R Houston (Ed.), Handbook of Research on Teacher Education. New York, NY:
Macmillan.
Bogdan, R & Biklen, S. K. (1992). Qualitative research for education: An introduction to theory
and methods (2nd Ed.). Boston: Allyn and Bacon.
Borko, H., Eisenhart, M., Brown, C. A, Underhill, R, Jones, D., & Agard, P. C. (1992).
Learning to teach hard mathematics: Do novice teachers and their instructors give up
too easily? Journal for Research in Mathematics Education, 23, 194-222.
Carroll, J. (1998). Developing a framework for viewing affective and knowledge factors in
teaching primary mathematics. In C. Kanes, M.Goos, & E. Warren (Eds.), Teaching
Mathematics in New Times (Proceedings of the 21st annual conference of the
Mathematics Education Research Group of Australasia, pp.137-144).
Brisbane:
MERGA.
Darling-Hammond, L. (1998). Teachers and teaching: Testing policy hypotheses from a
national commission report. Educational Researcher, 27(1), 5-15.
Department of Employment, Education and Training (1989). Discipline Review of Teacher
Education in Mathematics and Science (Vol. 1). Canberra: Australian Government
Printing Service.
Dewey, J. (1964). The relation of theory to practice in education. In R Archambault (Ed.),
John Dewey on education. Chicago, IL: University of Chicago Press.
Feiman-Nemser, S., & Buchmann, M. (1986). The first year of teacher preparation:
Transition to pedagogical thinking. Journal of Curriculum Studies, 18, 239-256.
Floden, R E., McDiarmid, G. W., & Wiemers, N. (1990). Learning about mathematics in
elementary methods courses (Research Report 90-1). East Lansing, MI: Michigan State
University, National Center for Research on Teacher Learning.
Foss, D. H. & Kleinsasser, R (1995, April). Preservice Teachers and a methods instructor:
Consensus, contrast and conflict. Paper presented at the annual meeting of the American
Educational Research Association, San Francisco, CA
Foss, D. H., & Kleinsasser, R (1996). Preservice elementary teachers' views of pedagogical
and mathematical content knowledge. Teaching and Teacher Education, 12,429-442.
Kennedy, M. M. (1990). Trends and issues in teachers' subject matter knowledge (Trends and
Issues Paper No.1). Washington, DC: ERIC Clearinghouse on Teacher Education.
(ERIC Document Reproduction Service No. ED322100)
Mayers, C. (1994). Mathematics and mathematics teaching: Changes in pre-service studentteachers' beliefs and attitudes. In G. Bell, B. Wright, N. Leeson, & J. Geake (Eds.),
Challenges in mathematics education: Constraints on construction (Proceedings of the 17th
annual conference of the Mathematics Education Research Group of Australasia,
Vol. 2, pp. 419-428). Lismore: MERGA
Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis (2nd ed.). Thousand
Oaks, CA: Sage.
Mosenthal, J. H., & Ball, D. L. (1992). Constructing new forms of teaching: Subject matter
knowledge in inservice teacher education. Journal of Teacher Education, 43, 347-356.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards
for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (1991). Professional standards for teaching
mathematics. Reston, VA: Author.
Quality Solutions and Research. (1997). NUD.IST. Melbourne: Latrobe University.
Teaching Mathematics: A Brightly Wrapped but Empty Gift Box
123
Schmidt, W., & Kennedy, M. (1990). Teachers' and teacher candidates' beliefs about subject
matter and about teaching responsibilities (Research Report 90-4). East Lansing, MI:
Michigan State University, National Center for Research on Teacher Learning.
Schuck, S. (1996a). Chains in primary teacher mathematics education courses: An analysis
of powerful constraints. Mathematics Education Research Journal, 8, 119-136.
Schuck, S. (1996b). Reflections on dilemmas and tensions in mathematics education
courses for student teachers. Asia Pacific Journal of Teacher Education, 24, 75 - 82.
Schuck, S. (1997). Using a research simulation to challenge prospective teachers' beliefs
about mathematics. Teaching and Teacher Education, 13,529-539. "
Schuck, S. (1998). Conversations of the three selves of the prospective primary
mathematics teacher. Teaching and Teacher EdlJ.cation, 14, 703-714.
Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational
Researcher, 15(2),4-14.
Strauss, A. L., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for
developing grounded theory (2nd ed.). Thousand Oaks, CA: Sage.
Sullivan, P. (1992, August). Students, classrooms, context, and the role of the teacher. Paper
presented at the 7th International Congress on Mathematical Education, Quebec,
Canada.
Sullivan, P., & Mousley, J. A. (1996). Describing and evaluating mathematics teaching. In
J. Mousley & P. Sullivan (Eds.), Learning about teaching (pp. 75-82). Adelaide: The
Australian Association of Mathematics Teachers.
Taplin, M. (1998). Preservice teachers' problem-solving processes. Mathematics Education
Research Journal, 10(3), 59-75.
Author
Sandy Schuck, Faculty of Education, University of Technology, Sydney, P.O. Box 222,
Lindfield NSW 2070.. E-mail: <sandy.schuck@uts.edu.au>.