Teachers’ reasoning about proportional relationships as variable parts
Sybilla Beckmann and Andrew Izsák1
The University of Georgia
1
We thank the Spencer Foundation and the University of Georgia for supporting
this research.
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Abstract: We present preliminary results from an ongoing study investigating how future
middle grades teachers use drawn models, such as strip diagrams and number lines, to
reason about quantities that vary together in a fixed ratio relationship, and how teachers’
understandings about multiplication, division, and fractions support and constrain that
reasoning. In this paper, we report on future teachers’ reasoning about ratios and
proportional relationships from the fixed numbers of variable parts perspective, a
perspective on proportional relationships that has been largely overlooked in past
research. Our findings indicate productive targets for instruction in middle grades teacher
education.
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Educational and scientific importance of the research
Ratios and proportional relationships are critical topics in elementary and secondary
mathematics (e.g., Kilpatrick, Swafford, & Findell, 2001; National Council of Teachers
of Mathematics, 1989, 2000; National Mathematics Advisory Panel, 2008), and a large
body of research amassed over the past 30 to 40 years on fractions, ratios, and
proportions (reviewed by Behr, Harel, Post, & Lesh, 1992; Lamon, 2007) has
demonstrated that these are some of the most challenging topics for students to learn. The
NCTM Curriculum and Evaluation Standards for School Mathematics (1989) described
proportionality to be “of such great importance that it merits whatever time and effort
must be expended to assure its careful development” (p. 82). The recent Framework for
K-12 Science Education (National Research Council, 2012) lists “Scale, proportion, and
quantity” as a crosscutting concept (p. 3) and stated that “Recognition of [ratio]
relationships among different quantities is a key step in forming mathematical models
that interpret scientific data” (p. 90). Still more recently, the National Center on
Education and the Economy (2013) identified lack of strong conceptual understanding of
middle school mathematics—especially arithmetic, ratios and proportions, and simple
equations—as the most important obstacle to readiness for community college, where the
bulk of vocational and technical education takes place and where many students complete
the first 2 years of a 4-year college baccalaureate program. Despite the importance of the
topic, there is little research on how teachers can develop robust understandings of ratios
and proportional relationships.
Traditional instruction in proportional relationships has emphasized applying rote
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procedures; but, current recommendations advise against the early introduction of
computational shortcuts, calling instead for students to reason about how quantities vary
together (e.g., Kilpatrick et al., 2001; Siegler et al., 2010). Thus, a robust understanding
of proportional relationships includes understanding and using multiplicative
relationships between two co-varying quantities and recognizing whether or not two covarying quantities remain in the same constant ratio. To help students reason about covariation, the Common Core State Standards (CCSS; Common Core State Standards
Initiative, 2010) explicitly call for two types of linear drawn models, strip (tape) diagrams
and number lines. Furthermore, this broader view of proportional reasoning, grounded in
quantities, can support CCSS mathematical practices such as making sense of problems,
reasoning quantitatively and abstractly, constructing viable arguments, and using double
number lines or strip diagrams strategically as tools.
Framework and research questions
We use a framework that is grounded in a mathematical analysis of proportional
relationships introduced by Beckmann and Izsák (2014), which concerns reasoning about
quantities—such a lengths and volumes—in terms of measurement units—such as
centimeters and cups. The framework emphasizes solving problems by reasoning directly
with quantities rather than by relying on memorized algorithms for computation. We also
draw on Vergnaud’s (1983, 1988, 1994) construct of the multiplicative conceptual field
that situates ratios and proportional relationships in a web of interrelated ideas including
whole number multiplication and division, fractions, ratios and proportions, linear
functions, and more.
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Beckmann and Izsák’s (2014) mathematical analysis of proportional relationships
identified two distinct perspectives. One perspective, the variable number of fixed
amounts perspective, has often been termed composed unit reasoning and has been
widely studied among children, (e.g., Lamon,1993; Lobato & Ellis, 2010). The second
perspective, termed fixed number of variable parts (or variable parts for short), has been
largely overlooked in past research on proportional relationships. The variable parts
perpsective is our focus in this paper.
The variable parts perspective is supported by strip diagrams (also known as tape
diagrams), such as in Figure 1, which show how quantities are related. Students can solve
proportion problems by reasoning about how the quantities depicted in the strip diagram
are related and expressing those relationships in terms of multiplication and divison.
To make Fruit Punch, mix peach juice and grape juice in the ratio 3 to 2.
How much grape juice should you mix with 5 cups peach juice?
(a)
peach
juice:
grape
juice:
5 cups
? cups
5 cups ÷ 3 parts =
5/3 cups per part
2 partstDVQTQFSQBSU
= 10/3 cups
peach
juice:
grape
juice:
Iterating and partitioning
5 cups
(b)
2/3t 5 cups =
10/3 cups
t2/3
2/3t P = G
? cups
Multiplicative comparison
Figure 1: Using strip diagrams and the variable parts perspective to solve a proportion.
(a) Partitioning and iterating. (b) Multiplicative comparison.
Our research questions included the following:
1. What opportunities and challenges do future teachers face when learning to use drawn
representations, such as strip diagrams, to support reasoning about ratios and
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proportional relationships in terms of quantities from the variable parts perspective?
2. What opportunities and challenges do future teachers face when using relevant
knowledge, such as understandings of multiplication and division, to make sense of
ratios and proportional relationships in terms of quantities from the variable parts
perspective?
Methods and data sources
In Fall 2012, we recruited four pairs of future middle grades teachers (Grades 4–8) and
four pairs of future secondary teachers (Grades 6–12) from preparation programs at one
large public university in the Southeast. Both groups were being prepared to teach the
grades in which ratios and proportions are typically taught. Courses for each program
developed quantitative meanings for multiplication and division, fraction arithmetic, and
the two perspectives on ratios and proportional relationships discussed above.
We recruited future teachers from courses we taught. We conducted two semi-structured
hour-long interviews with each pair, and participants were paid $25 for each hour. The
first interview examined how the future teachers determined whether relationships
between quantities described in story problems were or were not proportional. The
second interview, the focus of this paper, concentrated on reasoning with strip diagrams
about mixture problems from the variable parts perspective.
The interview tasks were presented on paper. We asked the future teachers to read the
tasks and then work together while explaining their reasoning out loud. Occasionally we
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asked clarifying questions. We recorded each interview using two video cameras, one to
capture the interviewer and participants sitting around a table together and one to capture
close ups of their written work and hand gestures they made towards their diagrams as
they explained their thinking. We combined the two files into a single synchronized file
and transcribed the interviews verbatim for analysis. We also collected all of the
participants’ written work. We took multiple passes through the data, reviewing the
transcripts side-by-side with the video. We attended to talk, gesture, and inscription for
evidence of what the preservice teachers were thinking.
Results
Our research revealed that the variable parts perspective on proportional relationships is
accessible to future teachers and provides them with a way to discuss and reason about
how quantities vary together in a proportional relationship. It also offers teacher
educators a window into future teachers’ quantitative understandings of multiplication,
division, and proportional relationships and reveals important foci for instruction on such
quantitative understandings.
All the prospective middle grades teachers we studied were able to use the variable parts
perspective to describe and reason about co-varying quantities in a proportional
relationship. But we also found considerable variability in whether and how the teachers
reasoned about quantities, especially with division. For example, in a situation in which
65 grams of substance is represented in a strip diagram with 12 equal parts separated into
a 5-part strip and a 7-part strip, it was not obvious to all teachers that the amount
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represented by one part could be found by dividing 65 by 12, perhaps because the two
strips represented different ingredients. We also found variability in whether and how
future teachers used strip diagrams to support their reasoning and interpret numerical
calculations. In some cases, teachers provided careful quantitative arguments about the
amounts represented in strip diagrams, justified expressions and equations in terms of the
diagrams, and attended closely to measurement units and other units. But in other cases,
especially in cases when fractions were involved, teachers used strip diagrams more
schematically, as a way to organize numbers and calculations, or did not refer to a strip
diagram at all to justify an expression or equation.
We found that reasoning about co-variation of quantities in a context where the
distributive property applies, as in the following example, was especially challenging for
future teachers.
Example task: To make 14-karat gold jewelry, people mix pure gold with
another metal such as copper to make a mixture that we’ll call “jewelry-gold.”
Jewelry-gold is made by mixing pure gold with copper in the ratio 7 to 5. There
was some jewelry-gold that was made by mixing pure gold and copper in a ratio 7
to 5. Then another 4 grams of pure gold was added to the jewelry-gold mixture.
What can you say about this new mixture?
Follow-up questions:
(a) Another student said that the new mixture is in the ratio 11 to 5. Do you
agree?
(b) If you want to keep the pure gold to copper mixture in the same 7 to 5
ratio, then how much copper would you need to add?
(c) Does it depend on how much pure gold and copper were in the mixture to
start with?
This task produced extended discussions about solution methods as most pairs struggled
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to solve or explain parts b and c of the task. Pairs often paid close attention to nuanced
differences in each other's quantitative reasoning, such as by distinguishing the
situational meaning of 5/7 times 4 from 4 times 5/7. Several pairs produced creative
drawn models to explain their thinking, sometimes adapting strip diagrams to fit their
reasoning. Only one pair invoked the distributive property explicitly. None of the pairs
treated the parts as variable.
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