· Mathematics Education Research ]oumal
1993, Vol. 5, No. 1,50-59
Connectedness in Teaching Equivalent Algebraic
Expressions: Novice Versus Expert Teachers
RuhamaEven
Dina Tirosh
The Weizmann Institute of Science
Tel Aviv University
Naomi Robinson
The Weizmann Institute of Science
This study examines differences in connectedness in instruction between two novice
teachers and an expert teacher. Three types of data related to lessons on equivalent
algebraic expressions were collected: lesson plans, lesson observations, and
post-lesson interviews. Although connectedness is an important characteristic of
mathematics teaching and learning, only the expert teacher used both lesson and
content connections to guide her teaching. Differences in the teachers' views and
uses of connectedness in instruction are discussed and illustrated.
Introduction
During the last decade the study of expertise in teaching has received much
attention (Berliner, 1986; Borko & Livingston, 1989). In the realm of mathematics,
the focus so far has been merely on expertise in the teaching of arithmetic
(Lampert, 1988; Leinhardt, 1989). The present study is aimed at identifying
dimensions of expertise in the teaching of a specific topic in algebra: equivalent
algebraic expressions.
The above studies identified several dimensions in which competency
differences between experts and novices are likely to occur, including planning
actions (agendas), managing action systems, and building explanations of
mathematical materials. With respect to planning actions, Leinhardt (1989) found
fundamental differences between experts and novices in the levels of
connectedness of their agendas:
The expert teachers always started their planning statements by telling what they had
done the day before, whereas none of the novices did so .., expert teachers saw lessons
as connected and tied together ... their agendas were richer in detail, in connectedness.
(p.64)
Rich connections constitute an important characteristic of mathematical
knowledge. This observation has two facets: mathematical and pedagogical. The
mathematical facet is related to seeing the discipline of mathematics as the study of
patterns and relationships (see, for example, Australian Education Council, 1991;
Dienes, 1960; Hoffman, 1989; Moore, 1903/1926; National Research Council, 1989;
Steen, 1990). The making of connections is recognised as a crucial element:
Mathematics involves observing, representing, and investigating patterns and
relationships in social and physical phenomena and between mathematical objects
themselves (Australian Education Council, 1991, p. 4).
Connectedness in Teaching Algebra: Novice vs Expert
51
The pedagogical facet is related to a conception of the nature of meaningfulleaming
of mathematics, in which the idea of making connections is no less central (Even,
1990; Hiebert & Lefevre, 1986; Lampert, 1988; Moore, 1903/1926; National Council
of Teachers of Mathematics, 1989, 1991; Steen, 1990). According to this view, one
cannot understand a mathematical concept in isolation. Interconnections between
concepts, representations, topics and procedures create insights which allow a
better, deeper, more powerful and more complete understanding. Meaning is
generated when relations between units of knowledge are recognised or created
(Hiebert & Lefevre, 1986).
One main role of a mathematics teacher is to help the learner construct
understandings of the subject matter. Creating classrooms in which the making of
connections is emphasised seems essential for this purpose. In such classrooms, the
U.S. National Council of Teachers of Mathematics (1989) envisaged that
ideas flow naturally from one lesson to another, rather than each lesson being
restricted to a narrow objective. Lessons frequently extend over several days so that
connections can be explored, discussed, and generalised. Once introduced, a topic is
used throughout the mathematics program. Teachers seize opportunities that arise
from classroom situations to relate different areas and uses of mathematics. (p. 32)
The study reported in this paper is part of a larger study on the multifaceted nature
of connectedness in teaching equivalent algebraic expressions. The larger study
focuses on two kinds of interrelated connections in teaching mathematics:
(a) connections in the representation of the subject matter (i.e., connections in
relation to what is learnt); and (b) connections with students' ways of thinking (i.e.,
in relation to the learner). In this paper we focus on the first type of connections,
and report findings related to differences between expert and novice teachers with
respect to connections in the representation of the subject matter (i.e., connections
in relation to what is learnt). These connections have two components:
1. Content connections across various concepts, representations, topics and
procedures.
2. Lesson connections within different lesson segments and series of lessons.
Method
Participants
Three teachers participated in this study. Two were novice teachers, both in
their second year of teaching. The other teacher was considered to be an "expert."
She had more than 15 years of experience and had acquired a reputation of being
an "excellent teacher" among students, colleagues, and experts in mathematics
education. All three teachers (the expert and the two novices) used a traditional
approach to the teaching of algebra-one that is common in Israeli schools. Their
lessons were teacher-centred with no emphasis on students' 'investigations and
communications.
52
Even, Tirosh & Robinson
Data Collection
The collection of data took place during separate lessons given by the expert
and novice teachers on the same teaching segment: the first three consecutive
lessons on equivalent algebraic expressions (open-phrases) in Grade seven. Three
types of data were collected:
1. Lesson plans-each teacher was asked to submit a lesson plan, before each
lesson, either in writing or on audiotape.
2. Observations-all lessons were observed by one of the researchers; field
notes were taken during observations and were supplemented by audiotapes.
3. Post-lesson interviews-semi-structured interviews after each lesson aimed
at examining teachers' reflections, and clarifying incidents or episodes from plans
or observations.
"
All interviews were audiotaped and transcribed.
Findings
Major differences in connectedness with respect to planning, teaching and
post-lesson reflections were observed between the novice and the expert teachers.
The following sections illustrate and discuss these differences.
The Expert
Planning and teaching. Connected, organised activities were evident throughout
the expert's planning of the lessons as well as in her teaching. Lesson segments
were connected with previous and following ones. For example, in order to link
together the first lesson on algebraic expressions with a previous lesson on
substitutions, she used the two previously discussed expressions: 7t+1 and
8t - (t - 1). In her lesson plan she wrote: "keep the two algebraic expressions in the
comer of the blackboard-we will deal with them later [italics added]." In doing so,
she planned to connect the new lesson with the previous one while introducing a
new topic through a context which was familiar and interesting to the students. She
then continued to use the same context for a series of lessons.
Another example of lesson connections between two lesson segments is taken
from an in-class observation. The teacher planned to work on equivalency of
expressions that included absolute value and algebraic fractions. The students
were first asked to determine whether the two algebraic expressions 2x and I x I + x
are equivalent. After concluding that they were not, the teacher moved to the next
stage of the lesson and asked: "Are the inverse expressions of non-equivalent
algebraic expressions equivalent?" She thus connected the two activities by
suggesting that they check the inverses of the two former algebraic expressions:
1
2x
and
1
Ixl +x
Connections were found not only in the expert's development of the lesson
Connectedness in Teaching Algebra: Novice vs Expert
53
towards the main objective (lesson connections) but also across various topics
(content connections). For example/ in a lesson plan about simplifying equivalent
algebraic expressions she wrote:
Let's look at the algebraic expression -2a.
a) What is the domain?
When are we going to get a negative number? Positive? Zero?
b) What is the relationship between the algebraic expressions -2a and 2a?
That means the algebraic expressions are ...
Domains/ substitutions and opposite algebraic expressions were not the main topic
of the lesson-they had been dealt with in previous lessons. Nevertheless/ the
expert teacher planned to make connections between these topics and the lesson's
main objective/ and then executed her plan. These content connections were made
fluently throughout the lessons; the students seemed to feel comfortable with this
and had no difficulty coming back to the main topic of the lesson.
The expert teacher not only followed her lesson plan but used opportunities
during the lessons to make unplanned content and lesson connections. For
example/ she planned a seat-work assignment for the students to check whether
the two algebraic expressions 1 + l/x and (x+l)/xare equivalent. During the lesson
she changed her plan (probably because she realised that students might have
difficulties) and worked on this assignment with the entire class.
The following episode illustrates how this teacher exploited opportunities to
make connections in the representation of the subject matter. Content connections
were made between specific lesson content and the more general issue of the role of
counter-example in mathematics; lesson connections occurred when algebra was
connected to numbers by using simple fractions to help students understand
operations on algebraic fractions. The teacher wrote the two expressions on the
blackboard/ and the following dialogue ensued:
T(eacher): What is the substitution set?
5(tudent)l: All numbers except O.
T: Are the two expressions equivalent?
52: No.
T: Can you give an example?
52: 4.
T: Well ...
52: I got confused.
53: They are equivalent.
T: Why?
53: Because the substitution sets are the same.
T: Does it mean that they are equivalent?
54: I can't find a counter-example.
T: If I can't find a counter-example, does it mean that the expressions are
equivalent? [Pause, no reaction]. Can I check all the examples?
55: No.
56: -2 is a counter-example.
[Students checked the substitution of -2 and realised they got the same result.)
57: I know that these two expressions are equivalent because I can bring them to
the same form.
54
Even, Tirosh & Robinson
The teacher confirmed the strategy suggested by the last student. Then, in order to
show that (x+1)/x can be written in the form 1 + l/x, she used numbers, a more
familiar context to the students than algebra (apparently in order to allow the
fluency of the lesson). She reminded them that (4+3)/2 can be written as 4/2 + 3/2,
and then connected the operation on numbers with algebra:
x+ 1
xlI
--=-+-=1+x
x x
x
Reflection. When reflecting on her teaching during the interviews, the expert
teacher said, without prompting, that she considered the making of connections
between different stages of the lessons and between different topics to be very
important, and deliberately planned to make them in her teaching. She consistently
mentioned, when reflecting on each of the lessons, the importance of
connectedness:
Every lesson is connected to a previous lesson either through the topic, if possible, or
through an activity they [the students] did before. I never start a new topic without
connecting it to a previous one. (First interview)
It is my objective to show that there are connections between the topics. (Second
interview)
[We should attempt] to show that everything is connected. (Third interview)
The Novice Teachers
Planning and teaching. While plarming the lessons and actually teaching them,
both novice teachers did not seem to acknowledge the importance of
connectedness. The different lesson segIl1.ents were unconnected and there were
almost no connections with previously learnt materials. The following example
from the first novice teacher's lesson plan illustrates this.
The students had just learned the definition of equivalent algebraic
expressions, namely, that the substitution of any number in the two expressions
gives the same result. The teacher had also shown them how to check two or three
numbers. He concluded (mistakenly) that, if the results are the same, then the
algebraic expressions are equivalent. For the following lesson, the teacher planned
to teach "another method" for checking equivalency, based on reducing algebraic
fractions and the distributive law. In his plan, the teacher made no connection with
the. "method" learnt in the previous lesson, nor did he explain why the new
method was appropriate or needed. Moreover, he started with the definition of
equivalency and immediately moved, without any connection, to the "new
method." The following is the first part of his lesson plan:
Let's remind ourselves what equivalent algebraic expressions are.
Repeat the definition.
And now let's check if the algebraic expressions are equivalent
m+6,
Are they equivalent?
2 (m + 6)
2
Connectedness in Teaching Algebra: Novice vs Expert
55
Hint: We can open the parentheses and try to reach the smallest expression possible.
At the end we reach the open phrase m + 6.
That means that we got from one algebraic expression to the other.
Let's take another example: the algebraic expressions 4a, a + 3a. Are they equivalent?
Yes-because we can add (collect terms) 4a = 3a + a.
We used the distributive property.
'a' has the coefficient 1.
3a has the coefficient 3.
We take' a' out and get (1 +3) . a.
We see that we used the distributive law. What did we use in the first example?reducing.
Therefore, we can show that algebraic expressions are equivalent by using rules and
conventions of number operations.
Moreover, the ,novice teachers seemed to be "goal-oriented," trying to reach the
objective of the lesson without paying attention to possible connections and to the
processes that lead to this objective.
The following episode from an in-class observation illustrates this. Novice 2
had written these algebraic expressions on the board:
4a+3
3a +6 +5a
2
Substitute a = 1/2.
Sl: You get the same result.
T:
T: Are the algebraic expressions equivalent?
S2: No, because we substituted only one number.
Sl: Yes.
S3: It is impossible to know. We need all the numbers.
S4: One example is not enough.
T: We can conclude-it is difficult to substitute numbers in a complicated
algebraic expression and therefore we should find a simpler equivalent
algebraic expression.
Although the substitution of a = 1/2 in the two given algebraic expressions can
lead naturally to the conclusion that "we should find a simpler equivalent
algebraic expression," this was by no means the kind of conclusion appropriate to
this discussion.
Usually, the novice teachers did not use opportunities that arose during the
lessons to make content connections between the main topic of the lesson and
previously learnt topics. For example, Novice 1 wrote the two algebraic
expressions 2a + 10 and 2(a + 5) on the board and asked the students to substitute
the numbers 4 and .,--5. One student then objected: "You are not allowed to
substitute -5." Although it was clear that this student, and others, were confused
about substitution sets, the teacher ignored his response and did not use this
opportunity to make content connections to improve students' understanding.
Instead, he moved on according to his plan.
Reflection. Neither of the novice teachers emphasised the making of connections
in planning and teaching. They differed, however, in the way they reflected on this
Even, Tirosh & Robinson
56
issue when prob~d by the interviewer. Novice 1 explicitly objected to the making of
content connections and said several times in his interviews that the main objective
of the lesson was the most important issue and therefore that "it is a waste of time"
to touch upon other topics. Otherwise, he claimed, he would never have enough
time to cover the topic of the lesson. In reference to lesson connections, to which
naturally he did not object, he responded to an interviewer's question by stating
that "there were no 'jumps' between different segments in my lesson."
In contrast, Novice 2 said that he had not planned to make content connections
to other topics because it had not occurred to him. Yet, in retrospect he said: "Now
I think that it could be important and interesting." He also felt that making this
kind of connection would be in conflict with "covering" the material. At the end of
the interview he asked the interviewer to assist him in connecting a particular topic
with other related mathematical topics (i.e., implementing the idea of making
content connections). This novice teacher was aware of "jumps" that occurred
during the lesson and explicitly regretted his lack of ability to cope with and to
integrate unexpected students' responses and questions into a fluent, coherent
lesson.
Table 1 summarises novice versus expert characterisation of connectedness
related to the three components: planning the lesson, teaching, and post-lesson
reflections.
Table 1
Differences in Connectedness Between Novice and Expert Teachers
Expert
Planning
Teaching
Reflections
Novices
Horizontal connections: a major
factor taken into account
No content connections
.qlentioned
Lesson connections emphasised
Segmented planning
Horizontal connections
observed at least six times per
lesson
Almost no content connections·
observed
Emphasis on lesson connections
observed
Weak lesson connections
observed
Exploited opportunities to make
connections
Neglected opportunities to make
connections
Connectedness mentioned as a'
major goal of instruction
Novice 1 argued that there is
neither time nor need for content
connections.
Novice 2 conceived both lesson
and content connections as
important but difficult to perform
Connectedness in Teaching Algebra: Novice vs Expert
57
Discussion
The teachers in this study-the novices as well as the expert-taught algebra in
a traditional way. They portrayed algebra as rigorous, formal and non-concrete.
None of the teachers, for example, used spatial arrangements to help students see
that the two expressions must be _equivalent, and to give meaning to algebraic
manipulation (Australian Education Council, 1991). They also, traditionally, did
most of the talking, modelling and explaining themselves. Still, even in this
traditional way of teaching, our initial exploration suggests that the participant
expert teacher largely differed from the novices in the emphasis she gave to lesson
and content connections in her planning and teaching of equivalent algebraic
expressions, as well as in her reflections on her own lessons. As we saw in this
study, the expert considered the issue of connectedness to be very important. She
thoughtfully connected between different lesson segments and series of lessons.
Content connections between the lessons' main topics and previously learnt
materials were an integral component of each lesson.
The novice teachers, on the other hand, did not emphasise connectedness in
their lesson plans and teaching. They tended to adhere to their rather segmented
lesson plans. During the actual teaching they were inattentive to the students,
ignoring what was happening in the lesson, and drew conclusions that suited their
original lesson plans but bore no connection to what went on in the classrooms. By
and large, due both to their fragmented planning and their insensitivity to
students' reactions, the novices' lessons lacked lesson connections. Moreover, the
planning and teaching of both novice teachers were characterised by a lack of
content connections. Yet the two novices differed in their level of appreciation of
content connections. Although one strongly objected to the idea of including such
connections in his teaching, the other considered them important.
When trying to reveal possible reasons for the substantial differences in the
occurrence of connectedness between expert and novice teachers, the first factor
that comes to mind is subject-matter knowledge. Mathematical knowledge
includes both knowledge of, and about, mathematics (e.g., Ball, 1991). Knowledge
of mathematics refers to understandings of particular topics, procedures and
concepts, and the relationships among them (Hiebert & Lefevre, 1986; Skemp,
1978)-this is usually referred to as subject matter knowledge. Knowledge about
mathematics is a more general knowledge about the discipline which guides the
construction and use of knowledge of mathematics. Knowledge about mathematics
includes understanding about the nature of mathematics, that is the ways, means
and processes by which truths are established as well as the relative centrality of
different ideas (Ball, 1991; Lampert, 1988; Schoenfeld, 1988; Shulman, 1986;
Thompson, 1984).
Knowledge of mathematics interacts with knowledge about mathematics.
Looking at mathematical knowledge in this way, the term "content connections"
acquires a more complex meaning than it seems to have at first glance. Content
connections include not only connections between different concepts,
representations, topics and procedures, but also between knowledge of
nlathematics and knowledge about mathematics. Powerful content connections are
those that are made between and within important concepts and central ideas in
58
Even, Tirosh & Robinson
mathematics. In order to make these connections in their teaching, teachers need to
have a thorough and comprehensive knowledge of and about mathematics. Thus,
subject matter preparation that emphasises connectedness empowers teachers to
make sensible instructional decisions related to content connections.
Although a key role of content connections is to deepen students'
understanding of subject matter, lesson connections have a· more pedagogical
function since they enable the teacher to develop a fluent and coherent lesson or
series of lessons in a way appropriate for the students. In fact, pedagogical content
knowledge seems to be another important factor which contributes to the
differences in connectedness of instruction between expert and novice teachers.
Shulman (1986) describes this kind of knowledge as (a) knowing those ways of
representing and formulating the subject-matter which make it comprehensible to
others; (b) understanding what makes the learning of specific topics easy or
difficult; and (c) knowing the conceptions and preconceptions that students of
different ages and backgrounds bring with them to the learning.
Awareness of Connections and the Professional Development of Teachers
Novices' lack of ability to make lesson connections could be explained, at least
in part, by a lack of knowledge and understanding of students' conceptions and
ways of thinking. Not being able to anticipate and refer to students' difficulties or
novel ideas can seriously hinder the ability of teachers to plan and execute
coherent, connected lessons. It is recommended that the importance of making
lesson connections become an integral part of the development of prospective and
practising teachers' understanding of students' ways of thinking about
mathematics.
The development of both the mathematical and pedagogical content
knowledge of prospective and novice teachers has the potential to raise their
awareness of connections, and to improve their ability to make knowledgeable
decisions about how and when to make connections. It is therefore suggested that
teacher educators examine means by which novice teachers could be guided to
view connectedness as an important characteristic of teaching. Further, we need to
find adequate ways to help teachers emphasise both lesson and content
connections in their teaching. One such way is to present case studies of expert
teachers (Shulman, 1986).
This study is embedded in the expert-novice paradigm which aims at mapping
the salient differences between novices and experts. Much like other studies on
expertise in domain-specific knowledge, we used this paradigm as a means of
characterising teachers' dimensions of expertise. Unlike many other studies, our
study examines differences between expert and novice teachers in a specific
context: equivalent algebraic expressions. Recent trends of analysis of subjectmatter knowledge for teaching have started to concentrate on specific content areas
(Even, 1990). These trends demonstrate that insight into what constitutes expert
teaching can be gained by dealing with specific content domains. We believe that
finding how expert teachers differ from novices in connectedness in instruction of a
particular content might contribute to our understanding of the essence of good
instruction, as well as guide us in developing ways to help novice teachers become
better teachers.
Connectedness in Teaching Algebra: Novice vs Expert
59
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Authors
Ruhama Even and Naomi Robinson, Department of Science Teaching, School of Education
The Weizmann Institute of Science, Tel Aviv University, Rehovot, 76100, Israel.
Dina Tirosh, School of Education, Tel Aviv University, Ramat Aviv, 69978, Israel