Vol. 5
296
MULTIPLE MEANINGS IN MATHEMATICS: BENEATH THE SURFACE OF AREA
Samuel Otten
Michigan State University
[email protected]
Beth A. Herbel-Eisenmann
Michigan State University
[email protected]
This study applied thematic discourse analysis (Lemke, 1990) to a section of a middle school
lesson focused on the relationship between the area of parallelograms and rectangles. This
analysis provides a way to show the structure of the semantic relations between mathematical
terms, shedding light on points of convergence and divergence between parallelograms and
rectangles that can be, and often are, used for instructional purposes. Points of teacher dialogue
where the semantic relations between mathematical terms might have been unclear to students
are identified and particular attention is given to subtle shifts in the meanings of the terms base
and height.
Background
Mathematical language presents several significant challenges to students (Schleppegrell,
2007). For example, mathematical terms are used with a different level of precision than terms in
everyday language, in part because a mathematical definition provides both necessary and
sufficient information about the term whereas the definition of an everyday term merely
describes its meaning. This distinction was taken further by Poincaré who pointed out that a
mathematical definition, rather than encapsulating an existing meaning, actually creates the
mathematical entity in question (Folina, 1992). Linguistic challenges also arise when words are
used differently inside mathematics classrooms than they are outside (Pimm, 1989; Thompson &
Rubenstein, 2000), as is the case with average, power, similar, right, and even the word or.
Another challenge of mathematical language, one that is especially relevant to the current study,
is that a single term is often used in ways that have subtly different meanings. A mathematician
using the word inverse may, depending on the context, be referring to an inverse function, an
inverse operation, or the multiplicative inverse of a group element. We are not arguing that such
uses of mathematical terms are inappropriate or undesirable because we recognize the value in a
compact, versatile language that mirrors the myriad connections between mathematical entities.
Rather, we are emphasizing that in the process of learning mathematics, which in part means
becoming fluent in its language and meaning systems (Chapman, 2003), overcoming such
obstacles is non-trivial for students. We contend that more detailed attention to such non-trivial
aspects of mathematics learning can be helpful to teachers and researchers, a point we return to
in the final section of this paper.
The fundamental geometric concepts of base and height provide another example of
mathematical terms that are used with subtly different meanings at different times. The purpose
of this study is to examine a classroom interaction that includes the terms base and height to see
whether and how this subtle shift in meaning manifests in the dialogue.
Theoretical Perspective
Michael Halliday’s (Halliday, 1978; Halliday & Matthiessen, 2003) theory of systemic
functional linguistics underlies the analytic methods we employ in this paper. A foundational
assumption of this theory is that context and language use are intimately related: context
influences language choice and language choice helps to construe context. Halliday described
Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31st annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.
Vol. 5
297
three metafunctions of language—ideational, interpersonal, and textual. Language is used to
make sense of experience and in so doing serves the ideational metafunction; that is, it is used to
give cues and clues regarding the meaning of what is being talked about. Language is also a
means for acting out the social relationships of those who are using the language, thus serving
the interpersonal metafunction. The textual metafunction refers to aspects of the organization of
the language itself. Unfortunately, we are unable in the present paper to provide a full description
of the theory, but we can make note of work in mathematics education that has taken up the
general ideas of systemic functional linguistics (e.g., Atweh, Bleicher, & Cooper, 1998;
Chapman, 2003; Morgan, 1998). For this study we focus specifically on the ideational
metafunction—that is, the content of mathematics—though we recognize the value of research
focused on other aspects of language and also recognize the artificial nature of isolating one
metafunction from the others. That said, we, along with others (e.g., Steinbring, Bussi, &
Sierpinska, 1998), feel that it is important to engage in forms of discourse analysis that focus on
mathematical content and meaning.
Lemke (1990), who applied systemic functional linguistics to transcripts of science lessons,
viewed language as ―a system of resources for making meaning‖ (p. ix). From this perspective,
language does not consist of mere grammar and vocabulary but is also seen as a semantic system
or a system of meaning that allows us to create ―webs of relationships‖ among and between
ideas. As Lemke highlights, there are many ways to talk about ideas, but the underlying meaning
or ―pattern of relationships of meanings, always stay the same‖ (1990, p. x). It is through the
patterns of semantic relations that meaning is construed. For example, a group of people may
talk about the leg of a table using a wide variety of particular words and sentence structures, but
the pattern across the particular instances will be the semantic relation that the leg is a part of the
table (a MERONYM/HOLONYM relation). Lemke (1990) articulated a method for thematic analysis,
described below, with the purpose of uncovering and examining such patterns. Examples of the
semantic relations we drew on in this paper are displayed in Table 1.
Table 1
Semantic Relations
Linguistic Terms
Description
MERONYM / HOLONYM
part of a whole
HYPONYM / HYPERNYM
subset of a set
EXTENT / ENTITY
space associated with an object
LOCATION / LOCATED
spatial relationship
SYNONYM / SYNONYM
equivalence relationship
Methodology
We employed Lemke’s thematic analysis method as a lens that would bring into focus the
mathematical content of a middle school lesson on area. The lesson comes from a sixth-grade
classroom in an urban Midwestern middle school. The teacher, Robert, is elementary certified
and had been teaching for seven years. Prior to the collection of the data presented here, Robert
had not been a member of any professional organizations and the textbooks he used (there were
several) can be described as conventional.
The analysis consisted of several stages. First, we selected the transcript excerpt from a larger
corpus of classroom observations.1 This selection was based on the pervasiveness of content
Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31st annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.
Vol. 5
298
terms used in the excerpt and the fact that a classroom observation of another teacher existed in
our corpus that also dealt with the topic of area. (An article involving a detailed analysis of both
of these transcripts is forthcoming). Second, we reviewed and filled in details (e.g., what ―this‖
refers to) of the transcript. Third, we identified the content words that were central to the lesson
and generated a clean map of the semantic relations between these terms. This clean map, which
Lemke referred to as an ideal map, was based on our own mathematical understandings of the
terms as well as the definitions of the terms presented in various mathematical textbooks. Fourth,
we went through the transcript excerpt line by line and, for each occurrence of the identified
content words, attempted to identify the semantic relation at play. Finally, we looked across the
semantic relations of the transcript for thematic patterns and developed a transcript map. The
processes of the analysis and the products of the analysis both contributed insights regarding the
content of the interaction.
Results
The mathematical terms that we identified for analysis were rectangle, parallelogram, area,
base, height, length, width, as well as the non-technical term bottom. There is a
HYPONYM/HYPERNYM relationship between rectangles and parallelograms since the former are
particular instances of the latter. Area is an EXTENT of a polygon because it is a measure of twodimensional space. The base of a rectangle or parallelogram is defined as one of its sides, thus
forming a MERONYM/HOLONYM relationship involving base. In particular, a base is by definition
a geometric ENTITY. The height of a rectangle or parallelogram, on the other hand, is defined as
the distance between the given base and the line containing the opposite side and so is a quantity.
This semantic difference is not always made explicit to students, as we shall see below, and may
be confounded when the phrase ―base and height‖ is used in a way that implies their
interchangeability. Furthermore, the term base can also refer to the EXTENT of the base segment
(as in the area formula ―base times height‖ which calls for the quantities), and the term height
can also refer to a line segment (often dotted) drawn between the base and the opposite side.
Thus base is defined as a MERONYM of a rectangle or parallelogram but can also refer to an
EXTENT (i.e., the length of the base segment). The term height, on the other hand, is defined as an
EXTENT but is also used to refer to a geometric ENTITY (e.g., the dotted line from the base to the
―top‖ of the parallelogram). This is captured in Figure 1, the clean map, by the fact that these
terms appear twice under rectangles and parallelograms (with the term-as-defined above the
other usage).
The semantic relations we have just described hold for both rectangles and parallelograms,
which we see in Figure 1 because the overall semantic structures of rectangles and
parallelograms with respect to area are quite similar. This structural similarity is one of the
reasons that Robert, in the excerpt below, chose to teach parallelogram area by appealing to prior
knowledge of rectangular area. There are, of course, important differences. For instance, length
and width are terms that are associated with rectangles but not generally with parallelograms.
Also, a side adjacent to the given base of a rectangle can be interpreted as a height (in the
ENTITY-sense) of the rectangle, forming a MERONYM/HYPONYM relationship between the height
and the rectangle. However, this relationship does not exist with non-rectangular parallelograms.
Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31st annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.
Vol. 5
299
Figure 1. A clean map wherein rectangle is a HYPONYM of parallelogram.
Turning now to the classroom transcript, we join Robert after he has reviewed the area
formula for rectangles and is about to transition to the development of the area formula for
parallelograms. We use boldfaced text to draw attention to the content words in the transcript
since these were a primary focus of our analysis.
1 Robert: OK, now it was important that you brought up parallel because the next one is
2 area of a parallelogram. You just described to me a parallelogram, which was this:
3 rectangles are parallelograms. [Student: Yeah, just turned.] Parallelograms have two
4 sets of parallel sides. That’s what a parallelogram is by definition. So what would you
5 guess the formula would be for finding the area of a parallelogram? Adam?
6
Adam: Length times width.
7 R: Length times width. OK, aren’t rectangles--, didn’t we just decide that rectangles
8 are parallelograms?
9
Ss: Yes. Yes we did.
10 R: We decided because I told you that (laughs).
11
S: Actually you decided.
12 R: OK, so using that, area would be equal to length times width. They call it a little
13 different. Instead of saying length times width, they say base times height. They say
14 base times height. [Ms: Face?] Base. We’re going to make a parallelogram from our
15 rectangle. On your picture of your rectangle I want you to make this…a diagonal line
16 like this. [Draws on overhead a segment from the upper right corner of the rectangle to
Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31st annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.
Vol. 5
300
the interior of the bottom side.] We’re going to make a parallelogram from a
rectangle. OK? Now, when I put this diagonal in what shape did I make?
Ss: Triangle.
R: Triangle. So what we’re going to do, we’re going to take this triangle off and we’re
going to put it on the other side [draws a congruent triangle on the left side]. We’re
going to put it over here. So if we have our rectangle, which is what we had before,
and we had our three centimeters by five centimeters. We cut this section off and we
just add it on to the other side. OK? So here is our diagonal.
S: That looks like a 3D figure.
S: That’s cool.
R: That’s how they’re getting your parallelogram.
From the first teacher turn (lines 1–5) we can identify several semantic relations. The first
and last sentences of this turn imply that parallelograms have an area which, as was established
in previous classroom interactions, is an EXTENT. Also, rectangles and parallelograms are related
by the phrase ―rectangles are parallelograms‖ in lines 3 and 7–8. By this the teacher means that
all rectangles are parallelograms (i.e., the set of rectangles is a HYPONYM of the set of
parallelograms), but the phrase is unclear because ―are‖ can also be used to mean equivalence
(e.g., rectangles are quadrilaterals with four right angles). Because of this ambiguity we will
denote the relationship using the original term ―are‖ in the transcript map (displayed at the
conclusion of the third portion of the transcript). We also see in lines 3–4 a definition of
parallelograms, with the implication being that a parallelogram is equivalent to or SYNONYMOUS
with a figure satisfying the definition (i.e., containing two parallel sides).
Using reasoning based on the relationship between rectangles and parallelograms, Robert
leads Adam to state the area formula of parallelogram, which the student describes as ―length
times width‖ (line 6). If we take ―rectangles are parallelograms‖ to mean (correctly) that the set
of rectangles is a subset of the set of parallelograms, then this reasoning about the area formula is
flawed; the subset relationship would imply that rectangles have the same area formula as
parallelograms but not the converse, as was assumed in the excerpt. If, on the other hand, we
take ―rectangles are parallelograms‖ to mean (incorrectly) that the two are equivalent, then their
area formulas would necessarily be equivalent as well. In lines 12–13, the teacher modifies
Adam’s statement to be base times height, noting that ―they call it a little different‖ than length
times width. This information from the teacher establishes a sort of equivalence between the
terms length and width and the terms base and height. Lines 14–27 add another semantic relation
between parallelogram and rectangle as we see that the former can be constructed from the
latter. A rectangle, however, is already a parallelogram so what is meant is that a non-rectangular
parallelogram can be constructed from a rectangle, but this is not explicit.
We continue in the transcript, picking up directly where we left off.
28 Robert: OK, now we said that the formulas were similar. We determined that because
29 we said that the rectangle was a parallelogram, so we said that the formulas are
30 similar. OK, this bottom section will be considered our base [shades in base on
31 overhead]. OK, base is like length, so it is similar. That would be our base [points to
32 the parallelogram on the projection screen]. Now, when we found perimeter of a
33 rectangle what sides did we add? Or, what did we add together? If we were going to
34 find the perimeter of this rectangle what would we add?
35
Students: The sides. The length.
36 R: Which would be what?
17
18
19
20
21
22
23
24
25
26
27
Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31st annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.
Vol. 5
301
37
38
39
40
41
42
S: Fifteen.
R: Five, five, three, and three [points to the sides of the rectangle], correct? [S: Yeah.]
So if we were going to find the perimeter we’d go around the outside. OK, this
diagonal here is not the height of it [points to slanted side of parallelogram on screen].
It is on the outside but it’s not the height. The height has to be perpendicular or at a
right angle to the base.
This excerpt begins with the statement that the area formulas are ―similar.‖ We see in line 31
that base is ―like‖ length and that they are also ―similar.‖ This is slightly weaker than the
previous semantic relation, which implied that base and length were different references for the
same thing. In line 30 the teacher states that the bottom of the parallelogram will be ―considered‖
the base. The semantic relation in this situation is not clear, but there is some sort of
identification or equivalence taking place between base and bottom. Furthermore, in lines 39–42
we see height for the first time and learn that it is not the slanted side of the parallelogram and so
not a MERONYM. Its relationship to the base is articulated as one of perpendicularity. Since
perpendicularity is a characteristic of actual entities and not quantities, this statement about the
height implies that it is an ENTITY and not a quantity.
The relations involving height will be further developed in the next excerpt. Before then,
however, we would like to point out the subtle semantic shifts occurring in lines 32–38. The
teacher begins by asking what would be added in the calculation of a rectangle’s perimeter,
hinting in line 32 that sides are involved. Students respond that the sides are added. Semantically,
however, it is not the sides themselves that are added but the EXTENTs or lengths of the sides.
This subtle distinction is even more clearly confounded in line 38 when Robert points to the
sides themselves but calls out their measurements. (Again, we do not wish to communicate that
such action is negative, but merely to illuminate the fact that two different semantic relations are
involved.)
We continue directly following line 42.
43 Robert: So if this was our parallelogram, the height would actually be this vertical
44 distance here between these two lines [draws vertical segment in the interior of the
45 parallelogram on the overhead]. OK, so this would be our height, from here down. Or
46 it’d be from here down [draws vertical segment in the exterior of the parallelogram].
47 That would be our height. The height is actually just the distance between those two
48 bases. So, what is the area of this figure here? Think about what we did with this
49 [points to the cut-off triangle]. What does it have to be?
50
Student: Fifteen.
51 R: Fifteen. Didn’t we just take this triangle and move it over here [points to rectangle]?
52
Ss: Yeah.
53 R: So doesn’t the area have to be the same? [Ss: Yes.] Yeah, so we know the base is
54 five and the height is three. These side lengths may be different. They may actually be
55 four or three point seven. They may be some other number. But the height has to be
56 perpendicular to the base. It’s got to be straight up and down for a parallelogram. OK?
We see in line 43 and again in line 47 that height is defined as a distance, which corresponds
with the definition used in the clean map in Figure 1. This explicit discussion of height does not,
however, correspond with the semantic relations in the previous excerpt in which height was
characterized as a geometric ENTITY rather than a quantity. The notion of height and base as
quantities reappears in line 54 when it is stated that ―the base is five and the height is three.‖ The
notion of height and base as entities reappears in lines 55–56 when Robert reminds the class that
Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31st annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.
Vol. 5
302
―the height has to be perpendicular to the base.‖ Thus this final turn is another example of the
subtle and implicit shift in semantics involving the terms base and height. Figure 2 contains a
transcript map based on all three excerpts discussed in this subsection.
Figure 2. A transcript map based on Robert’s excerpt.
Discussion
In this paper, we used thematic analysis to examine the semantic relations from a middle
school lesson on the area of parallelograms. In generating the clean map of the relations between
the pertinent mathematical terms (see Figure 1), we became acutely aware of the structural
similarities between the semantic relations of rectangular area and those of parallelogram area,
similarities that are often exploited in instruction as teachers and curriculum materials relate
parallelogram area to rectangles or vice versa. The clean map also allowed us to see where the
semantic differences lay between the two types of polygons, differences that could inform
instructional decisions. Moreover, the clean map process exposed the difference between base (a
geometric ENTITY) and height (an EXTENT) as mathematically defined, as well as the fact that the
same terms base and height are used to refer to both physical segments and the lengths of those
segments. Height has the additional distinction of having the semantic possibility of being a
MERONYM of a rectangle but never of a non-rectangular parallelogram. All of these subtleties and
possible points of student confusion appeared in the transcript from the sixth grade lesson. Base
often referred to a side but was also used as a number and plugged into the formula (e.g., line
54). Height was explicitly defined as a distance (e.g., line 43) but then was referred to as the
drawn-in segment and described as being perpendicular to the base (line 41); a property that only
makes sense in reference to a geometric ENTITY. One point the teacher did try to be clear about
was that the parallelogram’s height was not its ―diagonal‖ side. Perhaps this effort was a result of
his experience with students thinking that height, because it is a characteristic of a polygon or
because it seems interchangeable with base, is necessarily a MERONYM of that polygon. Indeed,
Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31st annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.
Vol. 5
303
all of these subtleties and implicit shifts in meaning involving height may be related to
documented difficulties with the concept (e.g., Gutierrez & Jaime, 1999).
In summary, we have looked closely at the ideational metafunction of language from a lesson
on area. We have seen that even the terrain of a topic such as parallelogram area, which appears
smooth from a distance, can contain many potential potholes and pitfalls. If, however,
researchers and teachers examine and come to better understand the structure and patterns of the
semantic relations between mathematical terms, they can make mathematical language, and thus
mathematics itself, more navigable for students.
Endnote
1. This data was collected as part of an NSF grant (#0347906) focusing on mathematics
classroom discourse (Second Author, PI). Any opinions, findings, and conclusions or
recommendations expressed in this article are those of the authors and do not necessarily reflect
the views of NSF. We would like to thank the teachers for allowing us to work in their
classrooms.
References
Atweh, B., Bleicher, R. E., & Cooper, T. J. (1998). The construction of the social context of
mathematics classrooms: A sociolinguistic analysis. Journal for Research in Mathematics
Education, 29, 63-82.
Chapman, A. P. (2003). Language practices in school mathematics: A social semiotic approach.
Lewiston, NY: Edwin Mellen.
Folina, J. (1992). Poincaré and the philosophy of mathematics. New York: St. Martin’s Press.
Gutierrez, A., & Jaime, A. (1999). Preservice teachers’ understanding of the concept of altitude
of a triangle. Journal of Mathematics Teacher Education, 2, 253-275.
Halliday, M. (1978). Language as social semiotic: The social interpretation of language and
meaning. Baltimore, MD: University Press.
Halliday, M., & Matthiessen, C. M. (2003). An introduction to functional grammar. Oxford, UK:
Oxford University Press.
Lemke, J. L. (1990). Talking science: Language, learning, and values. Norwood, NJ:
Greenwood Publishing.
Morgan, C. (1998). Writing mathematically: The discourse of investigation. Bristol, PA: Falmer
Press.
Pimm, D. (1989). Speaking mathematically: Communication in mathematics classrooms. New
York: Routledge.
Schleppegrell, M. (2007). The linguistic challenges of mathematics teaching and learning: A
research review. Reading and Writing Quarterly, 23, 139-159.
Steinbring, H., Bussi, M. G. B., & Sierpinska, A. (Eds.). (1998). Language and communication
in the mathematics classroom. Reston, VA: National Council of Teachers of Mathematics.
Thompson, D. R., & Rubenstein, R. N. (2000). Learning mathematics vocabulary: Potential
pitfalls and instructional strategies. Mathematics Teacher, 93, 568-574.
Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31st annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.