Conceptual Insights, Viable Arguments, and Technical Handles
Conceptual Insights, Viable Arguments, and Technical Handles
David Yopp
University of Idaho
Author Note
Correspondence concerning this paper should be addressed to David A. Yopp, Departments
of Mathematics and Curriculum and Instruction, University of Idaho, 313 Brink Hall, Moscow,
Idaho 83844, dyopp@uidaho.edu
Interactive Paper Session
Tuesday, April 8, 2014, 8:30-9:45 AM
Ernest N. Morial Convention Center, Room 206
NCTM Research Conference
New Orleans
April 7-9, 2014
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Conceptual Insights, Viable Arguments, and Technical Handles
Abstract
Findings from a generative study that develops a framework for describing the viability of the
arguments prospective elementary teachers articulate are reported. Identified are three types of
technical handles that appear important in articulating viable arguments for topics in a
mathematics course for prospective elementary teachers. Inappropriate and inadequate technical
handles are also noted.
Key words: argument, technical handles, conceptual insights
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Conceptual Insights, Viable Arguments, and Technical Handles
1. Perspectives or theoretical framework for the research
This research reports on findings of a generative study that developed a framework form
describing the viability of the arguments produced by prospective elementary teachers (PSTs)
articulate for number sense and operation topics found in the elementary and middle grades. I use
the term “articulated” argument because my perspective here is to analyze the final,
communicated argument and its viability from a technical skills point of view. This work is an
analysis of how the arguer culminates findings, observations, and insights into an argument.
Identify are three types of technical handles that appear important in articulating viable
arguments. Inappropriate and inadequate technical handles are also noted. The findings reconceptualize existing frameworks for constructive ways students handle data and insights
during argumentation (Raman, M., Sandefur, Birky, Campbell, & Somers, 2009; Sandefur,
Mason, Stylianides, & Watson, 2013)
1.1. Common Core and viable arguments
Toulmin’s (1958/2003) argument analysis scheme has been use to describe teachers’
mathematical arguments (e.g., Giannakoulias, Mastorides, Portari, & Zachariades, 2010;
Whitenack, Cavey, & Ellington, 2014). I find the model useful as an instructional framework for
guiding prospective elementary teachers toward responding appropriately to conjectures. To me,
“appropriate responses” are viable arguments for or against a mathematical conjecture/claim. My
description of “argument” is based on both Toulmin’s use of the term and an adaptation of
“viable argument” as described in CCSSM (2010). While CCSSM (2010) does not explicitly
define viable argument, I assume this term is used instead of “proof” to note that there are
acceptable argument types other than formal, mathematically logical ones. This assumption was
compelling to me as I developed my framework.
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Conceptual Insights, Viable Arguments, and Technical Handles
I focus on arguments that can be modeled sufficiently using the core of the argument as
described by Krummheur (2007). Such arguments have a claim, data, and a warrant. For me, this
core is said to be viable if it (1) expresses a clear, explicit, unambiguous, and appropriately
worded claim, (2) expresses support for that claim that involves acceptable data (foundation), (3)
expresses an acceptable warrant (narrative link) that links the data to the claim, and (4) identifies
the mathematics on which the argument relies. Acceptable data/foundations include examples,
diagrams, prior results, definitions, narrative descriptions, stories, etc., provided that the
representation of the data/foundation can be appealed to appropriately in the warrant. For
instance, examples are acceptable when crafting arguments for generalizations when they are
generic examples (Balacheff 1988; Sandefur, Mason, Stylianides, & Watson 2013; Yopp & Ely,
under review). Examples are considered generic examples provided they are used as referents to
illustrate objects, properties, and relationships shared by all possible examples. Viable examplebased arguments become possible when students generically appeal to those features expressed
in the examples when supporting their claims (Yopp & Ely, under review.).
Acceptable warrants (or narrative links) express how the data/foundation is used to support
the claim. If the claim is a generalization, then the warrant must express how the data is used to
represent all cases. In generic-example arguments, warrants must communicate how the
examples are appealed to in representing all cases and communicate how the properties
demonstrated in the examples are common to all plausible examples in the domain of the claim.
Criterion (4), identifies the mathematics on which the claim relies, also needs elaboration.
Mathematical claims are based on the meaning of the objects and operations involved in the
claims and their truth relies on these meanings. These “meanings” are determined by definition,
axioms, and theorems. Viable arguments must express these meanings, at least semantically.
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Conceptual Insights, Viable Arguments, and Technical Handles
Example-based arguments for existence claims, including those in CE arguments, fit into my
viable argument framework as follows: a candidate example(s) is presented as the
data/foundation, and a demonstration that the candidate(s) has the desired properties is presented
as the warrant. For counterexample arguments, demonstrating that the candidate has the desired
properties means showing that the candidate is in the domain of the generalization but does not
satisfy the conclusion. At times, “that the candidate is in the domain of the generalization” is
present without support and taken as self-evident.
1.2. Technical handles and conceptual insights
Raman, Sandefur, Birky, Campbell, & Somers (2009) suggest movements, not always found
in order, in creating a proof. The first is getting a key idea (a reinterpretation of that proposed in
Raman, 2003) that gives a sense of belief and understanding that may or may not provide clues
about how to write a proof. The second is a technical handle that gives a sense of “now I can
prove it” and a way to communicate the ideas behind the proof. The third is a culminating of the
argument into a standard form (relative to audience).
Sandefur et al (2013) recast the Raman et al framework and report “two important
components” (p. 328) to creating a proof as “(1) finding a conceptual insight (CI), i.e., a sense of
a structural relationship pertinent to the phenomenon of interest that indicates why the statement
is likely to be true, and (2) finding some technical handles (TH), i.e., ways of manipulating or
making use of the structural relations that support the conversion of the CI into acceptable
proofs” (p. 328).
I used these conceptions as a priori lens to analyze my data and found that I needed to
reinterpret them to represent my data. My analysis is different from the Raman et al and
Sandefur et al analysis in that I focus on only what is in the articulated argument as a product of
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Conceptual Insights, Viable Arguments, and Technical Handles
argumentation when I analysis the “handlings.” Also, my data arose from PSTs who examined a
conjecture and developed a claim, as opposed to proving or disproving a proposition given to
them as in the Sandefur et al and Raman et al studies. The distinction is subtle but important.
If I were to give students a mathematical statement to prove or disprove under a typical
didactic contract, then the student is to go through some process to develop an argument for that
statement or a counterexample for that statement. If I instead develop a standard in my classroom
that appropriate responses to claims are to include a claim (including an alternative claim with a
reduced domain), foundation, and narrative link, then the didactic contract has changed. Here,
the claim is as much a part of the articulated argument as is its support. Under this contract,
students are free to claim that the original conjecture is true or false or develop any number of
alternative claims. A PST who confirms a proper subset of the cases in the domain of the claim is
free to claim “the conjecture is true for these particular cases,” and is under no “contract” to
develop an argument for the entire set of cases to which the claim applies. This would represent
an “appropriate handling” of the empirical data in forming a claim. This makes the articulated
claim a type of handling of data and conceptual insights. This alone makes my analysis and
framework distinct from those developed by Sandefur et al and Raman et al. This focus helped
me draw clearer distinctions between CIs and THs and offers a description of what the arguer
was able to appropriately and viably express when culminating ideas, findings, and insights into
an argument product.
For this study, the term conceptual insights (CI) can refer to any one of the following:
developing a sense or belief based in pertinent mathematical structure that a claim is true or
false, developing a sense or belief based in pertinent mathematical structure about what might be
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Conceptual Insights, Viable Arguments, and Technical Handles
claimed (stated as true), or developing a sense or belief based in pertinent mathematical structure
about why a claim is true or false or what causes the claim to be true or false.
Through my narrowed focus on the articulated argument, I was able to identify three distinct
but related “handles.” Technical handles of type 1 (TH1) describe the articulated claim in
relation to the expressed data (foundation), CIs, or warrant (narrative link). Technical handles of
type 2 (TH2) describe how the data or CIs are expressed relative to the claim’s type (e.g.,
existence or generalization). Technical handles of type 3 (TH3) describe the expressed link
between the data or CIs and the claim (i.e., warrant). THs are first described without any
connotation of whether or not the handles are constructive. Adjectives (e.g., appropriate,
inappropriate, problematic, adequate, and inadequate) are applied to note a TH’s subtype and
potential for viable argumentation. This framework will be further exemplified in the results
section.
2. Methods, techniques, or modes of inquiry for the research
Research question:
How can we describe the viability of prospective teacher’s articulated arguments from a
“technical skills” point of view?
The research question was addressed using a generative study (Clement, 2000) of data
collected during a 10 week teaching experiment (Cobb and Steff, 1983) designed to improve
preservice elementary teachers’ ability to construct viable arguments. Teaching experiments
document the interaction between students and a learning trajectory, as well as the interaction
between students and the teacher/researcher. These environments are well suited for generative
studies that build models for how students might develop a particular mathematical practice and
document challenges to that development.
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Conceptual Insights, Viable Arguments, and Technical Handles
The preservice teachers were given the viable argument framework described in Section 1.1
as a standard for argumentation and were given practice using the framework. PSTs were asked
to investigate conjectures and respond with a viable arguments for approximately a one month
period. Explicit instruction was given about the differences between generalitaions and existence
statements, what constitutes a viable argument for each, and the mathematical practices of
conjecturing, testing, and revising claims and arguments as described in Lakatos (1976).
Students engaged in tasks collectively (groups of 4 or 5) both in class and by posting in an online
environmnet through asynchronous computer mediated communications. Students were
instructed that their response to tasks should articulate an argument and must express, at
minimum, one of the following argumentation features: a claim, data or a foundation, a warrant,
a backing, or a critique of another’s posted argument feature.
Analysis methods were akin to those described by Miles and Huberman (1994) in which the
analyst begins with a theoretical coding framework that is constantly compared to the data until a
model that fits the data emerges. From the perspective of a generative study (Clement, 2000), I
developed new elements or modifications to an existing theoretical model to explain data. A
cyclic process of analysis, refinement, and reanalysis was used to test the emerging framework
against the data (similar to the methods described in Sandefur et al., 2013). Such processes are
consistent with studies involving task-based interviews in which the goal is to revise models for
students thinking and developing thinking (Goldin, 2000). As conceptual themes emerged, the
themes were verified through triangulation with multiple data sources. For example, as PSTs’
posts were memoed, emerging themes were triangulated with task-based interview data to
confirm the technical handle codes.
3. Data Sources or Evidence for the Research
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Conceptual Insights, Viable Arguments, and Technical Handles
Data was collected from PSTs enrolled in a undergraduate mathematics content course for
elementary school teachers during two iterations of the experiment (21 PSTs in year 1, and 18
PSTs in year 2). Five sources of data were collected and used to develop the framework: (1)
students’ weekly posts in the online environment (2) teacher/researcher observations during
inclass work (3) student writen responses to inclass tasks (4) task-based, clinical interviews
(audio taped and transcribed) and (5) responses on paper-and-pencil assessments. This article
only reports data from the online tasks. Because in online enviroments researchers have less
control over the integretity of the data, all claims are about what the PSTs chose to present as
arguments, not on the source of their ideas. Subsets of the data were verified through cognitive
interviews with select PSTs to add to the integretity of the data and clarify ambiguties.
PSTs responded to a variety of tasks over a one-month period, but for this article, only
responses to the following tasks are reported:
Task 1: Maria’s claim: if a numbers is a multiple of 3, then it is also a multiple 6. Write a viable
argument for or against Maria’s claim.
Task 2: Jamal’s claim: if a numbers is a multiple of 6, then it is also a multiple 3. Write a viable
argument for or against Jamal’s claim.
Task 3: Sophia says that the sum of 3 consecutive counting numbers is divisible by 3. Isabella
says, “I think that the sum of 4 consecutive counting numbers is divisible by 4.” Write
exemplary responses, which include viable arguments, for or against the claims.
Task 4: Develop a viable argument for or against the claim for all natural numbers n ,
n2 + n + 41 is a prime number.
4. Results
4.1 Technical handles of type 1: stating appropriate claims
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Conceptual Insights, Viable Arguments, and Technical Handles
PSTs expressed appropriate TH1s when they communicated appropriately worded claims
that follow purdently from data or conceptual insights they presented. PSTs expressed
problematic claims we they worded the claim in a manner in which the meaning was ambiguous
or the claim was false when redrafted using formal mathematical quantifiers. PSTs expressed
inappropriate TH1s when they overgeneralized relative to the data presented.
A representative case of a problematic TH1 is found in Mary’s argument below. Mary offers
a claim that can be read as a generalization.
Mary’s online response to prompt 1:
Claim: Maria is incorrect in her claim; numbers that are multiples of 3 are not
multiples of 6 as well.
Foundation: 9\3=3[;] 9\6=?? although we would get a number from this, it
wouldn't be an even number.
Mary likely means there are multiples of 3 that are not multiples of 6 and is unaware that the
literal interpretation of her claim is that all multiples of 3 are not multiples of 6. Regardless of
her intent, there is a mismatch between the literal meaning of her claim and the foundation she
presents. I view the problematic claim as a reflection of her technical skill in stating appropriate
claims, not in her understanding of the situation. True, Mary likely has an inadequate knowledge
of the mathematical registry and canonical ways of stating claims, but I also acknowledge that
there are natural language ways of stating appropriate claims such as “some multiples of three
are multiples of 6.”
Several PSTs posted claims that use parts of the mathematical registry that are canonically
associated with a generalization but expressed other information in their posts indicating they
were presenting a counterexample. Ellen for example posted the claim “if a number is multiple
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Conceptual Insights, Viable Arguments, and Technical Handles
of 3 it is not a multiple of 6” then presented a collection of counterexamples.
Other times, PST’s inappropriate claims were a clear case of overgeneralizing based on the
data and CIs presented in the articulated argument. A representative inappropriate claim is found
in Margaret’s work below. Here, the PST overgeneralizes based on her data:
Claim: For all natural numbers n, n^2+n+41 will equal a prime number.
Data: 98^2+98+41=9743 –prime; 11^2+11+41=173 –prime; 43^2+43+41=1933 –
prime.
Margaret returned to the discussion after a counterexample was posted by a peer and
recanted her false claim. This provides evidence that she indeed overgeneralized. Yet,
even after her peers noted that the claim Margaret presented is false, some revised the
claim, and overgeneralized again by asserting that the expression is prime for all counting
numbers except the counterexamples observed.
Donna:
Since Ray has proven that this doesn’t work when n=41 [and n=412], maybe we
can rewrite the claim so it says something such as: Claim: n+n^2+41 will be a
prime number if n is not equal to 41 or 41^2. I’m not sure if there is anything else
that can be added to this point, but at least this claim covers what we have already
discussed.
My overarching analysis in this section is that all the PST whose work is presented
express a lack of technical skill in stating appropriately worded claims relative to the data
or CIs presented. If, for example, Margaret had the appropriate technical training, she
might “handle” her data more appropriately by stating, “ n2 + n + 41 ” is prime for these
three values.” I am not asserting that PSTs should not make conjectures, and even wild
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Conceptual Insights, Viable Arguments, and Technical Handles
guesses based on observerations, insights, and intuition. This is an important part of
mathematical thought. I am asserting that PSTs should be trained in making
unambiguous, appropriately worded claims, and that when it comes toime to culminate
ideas into an argument to be articulated, they should only make claims that their data and
CIs support and nothing more, unless perhaps they include qualifiers. Intuition-based
claims and conjectures can be presented in prose outside the argument structure.
4.2 Technical handles of type 2: presenting adequate foundations
Adequate TH2s were noted when PSTs expressed their data or CIs in a manner that
could be appealed to appropriately when developing a viable argument for their claims.
Adequacy of the TH2 is dependent on the type of claim presented (e.g., existence or
generalization). Inadequate TH2s were noted when preservice teachers failed to express
data or CI in a manner that can be appealed to appropriately when supporting the claim,
and this is most often found when a generalization is stated. This code is most useful
when the observer is confident about the type of claim being presented (i.e., existence or
generalization). At times the ambiguity forces the observer to note both an inappropriate
TH1 relative to the foundation and an inadequate TH2 relative to the claim. Without a
cognitive interview, such arguments are double coded with two possible statuses.
A representative case of an inadequate TH2 is found in Jody’s argument below.
Jody’s response to prompt 2:
Claim: If a number is a multiple of 6, then it is also a multiple of 3
Foundation: Multiples of 3: 6,9,12,15,18,21, ...Multiples of 6: 12,18,24,30,36
Narrative Link: This is true because 3 is a factor of six. It can be said for all; that
if A is a factor of B, then B will also be a multiple of A.
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Conceptual Insights, Viable Arguments, and Technical Handles
In his narrative link, Jody expresses a useful conceptual insight that 3 is a factor of 6, and yet his
foundation does not express this idea. His foundation is purely empirical and cannot be appealed
to generically. Jody need not use a variable to represent this insight. A generic example such as
24 = 6 ´ 4 = 3´ ( 2 ´ 3) would express the relevant structure adequately for generic appeal when
supporting the claim.
Other PSTs presented similar arguments with examples that demonstrated that 3 and 6 are
both factors of particular numbers (e.g., 24 = 4 ´ 6 and 24 = 3´ 8 ). These foundations fall short
of adequacy because they do not show how the factors 3 and 6 are related and do not show the
pertinent structure (3 is also factor of 6). This lack of structure in the foundation means we do
not see how the claim’s conditions (multiple of 3) is used to demonstrate the conclusion
(multiple of 6).
4.3 Technical handles of type 3: presenting adequate narrative links
An adequate TH3 is expressed when an PST vaiably links the data or CIs to the claim,
explaining how the data or insights are used to support the claim. An inadequate TH3 is observed
when a PST presents an appropriate TH1 and an adequate TH2 but fails to explain how the data
or CIs are used to support the claim or fails to explicitly discuss the mathematics on which the
claim relies.
A representative case is found in the work of Sarah who posts the following argument:
Claim: Sophia is correct by saying that the sum of 3 consecutive numbers are [sic]
divisible by 3.
Foundation: A+(A-1)+(A-2)=B[.] B is divisible by 3. Ex. 9+8+7=24[.] 24/3=8. There is
an infinite amount of situations where this is true.
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Conceptual Insights, Viable Arguments, and Technical Handles
Narrative: A+(A-1)+(A-2)=B always gives a number that can be divisible by 3. This
works every time because of the fact that you are just moving on the number line. There
are an infinite amount of situations stating that this is true.
Sarah’s claim is clearly a generalization (e.g., “infinite situations”), and one might argue that that
Sarah’s foundation is inadequate because she does not show how the representation of three
consecutive is used to express the conclusion (e.g., 3( A -1) ). But, Sarah might wish to present
such information in the narrative link. To be true to the foundation Sarah presents, I turn my
attention to her narrative link.
Although Sarah refers to her foundation in her narrative link, she does not appeal to it
explicitly. She appeals to “moving on the number line” in her narrative, and it isn’t clear how
“number line” is expressed in the foundation, if at all. Thus, Sarah expresses an inadequate TH3
because she does not explain how her foundation supports her claim.
In her second argument, Sarah presents another generalization with a similar foundation:
Generalization: The sum of 4 consecutive numbers is divisible by 2.
Foundation: A+(A-1)+(A-2)+(A-3)= B [.] B is divisible by 2[.] Ex. 4+3+2+1=10[.]
10/2=5[.]
Narrative: The sum will make 2 odd numbers, which then would be added and make one
even number. The sum of 4 consecutive numbers will be even which will make it
divisible by 2.
Here Sarah presents a conceptual insight in her narrative link (that the sum of four consecutive
numbers is equal to the sum of two odds), which the foundation does afford. This would require
cases (e.g., A even and A odd) and an adequate narrative link would have to appeal to this
explicitly. An adequate narrative link would also need to explain that four consecutive numbers
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Conceptual Insights, Viable Arguments, and Technical Handles
alternative between even and odd, and that the associative property is used to add the first two
and last two consecutives. Finally, explicit mention of the assumed prior result that an even plus
an odd is odd might be appropriate for this audience.
Again, a reader might argue that Sarah could/should demonstrate more structure in her
foundation (e.g., 3( A -1)). An analyst using Toulmin’s (1958/2003) full argumentation scheme
might construct an idealized argument from Sarah’s work that places some of the missing
features in the foundation and other information as backing (e.g., even plus an odd is odd). My
analysis here attempts to be true to the argument Sarah chooses to articulate. Because I find ways
to create a viable argument using the foundation Sarah presents, my analysis is that her TH3 is
inadequate based on the presented generalization and foundation.
As an aside, I wish to mention my views about the example’s Sarah presents in her
foundation. Some authors have suggested that students who desire examples to complement a
general argument might be expressing a lack of faith in the “proof” (e.g., Harel & Sowder,
1998), but this does not influence my analysis of Sarah’s handlings. I view the examples as
superfluous in Sarah’s case because there is no appeal to them in her narrative.
In sum, Sarah’s work expresses an inadequate technical skill in developing a coherent
foundations and narrative links. Sarah has not adequately explained how the expressions in her
foundations support her claims. Moreover, the mathematics on which the claims rely is not
clearly articulated. We wonder, for example, what mathematics is behind the insight “moving on
the number line” in the argument, if any.
5. Educational Importance of the Research
5.1 Putting the technical handles findings into an instructional framework
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Conceptual Insights, Viable Arguments, and Technical Handles
I am currently working implementing a teaching experiment that seeks to improve PSTs’
abilities to present viable arguments. The experiment includes explicit training on the
mathematical registry for stating claims and training on develop adequate foundations and
narrative links based on the types of claims stated. The framework has been useful in facilitating
student reflections on and critiques of their arguments. Below are arguments and descriptions of
their potential as tools for reflection and feedback using the framework.
Deborah’s response to task 1:
Claim[:] that Maria's claim could be correct.
Foundation[:] 24 is the multiple of 1,3,2, 12, 6, 8, 4, and 24
Narrative[:] I am understanding that a multiple can be a number if two different
factors are multiplied to get that multiple. So I can take a multiple of 3 and a
multiple of 6 and say that 18 is a multiple of 6 and of 3 and that 6 and 3 are
factors of 18. Not all multiples of 3s are going to work with 6, but a multiple is
what you get after multiplying a number by an integer. It would be all the factors
that equal a certain number for the multiple.
Deborah’s argument is not well connected across features. Maria’s original claim is a
generalization that multiples of 3 are also multiples of 6, and Deborah’s narrative link
expresses ideas about both conforming cases and counterexamples to Maria’s claim. Her
narrative link suggests the possibility of counterexamples, but none are given. Her
foundation presents one conforming example, and her narrative link presents another. My
first question for Deborah is whether or not she intents to make a generalization about a
large or infinite class of multiples of 3, or whether she wishes to make an existence
statement. If it is the former, I should guide Deborah toward an appropriately worded
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Conceptual Insights, Viable Arguments, and Technical Handles
generalization claim (e.g., every other multiple of 3 starting with 6 is also a multiple of 6)
and help her develop a representation that can express the structure for all cases. If it is
the later, I guide her toward stating an existence claim (either about conforming cases or
counterexamples) and guide her toward either placing both examples in the foundation
(there exist two…) or deleting one of the examples (there exists…).
Another PST, Amy offers a refutation to Isabella’s claim that appears to be two
arguments combined:
Claim: There are numbers of 3 that are multiples of 6. Every other multiple of 3
will be a multiple of 6 starting with 6.
Foundation: Multiple of 3: 3,6,9,12,15,18,21,24... Multiples of 6: 6,12,18,24
Narrative link: We can see that some multiples of 3 are multiples of 6. By
showing Maria that every other multiple of 3 is a multiple of 6 she can understand
there can be numbers that are multiples of both but it is not always true. Since 3 is
half of 6 then every other number will be a multiple of 6.
Amy’s first sentence is an existence claim that is supported by some of her examples in
the foundation and her first line in the narrative link that use the wording “some multiples
of 3.” Her second sentence in the claim is a generalization and her narrative has
pedagogical features. Amy’s narrative link ends with an observation that appears to be
the CI for the generalization. The focus here can be on how to represent the structure of
this CI in the foundation (e.g., 3´ (2n) = 6 ´ (n) ).
Amy’s argument can be used as an example for instruction that suggests developing
two separate arguments for refutations: a counterexample (CE) argument and an
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Conceptual Insights, Viable Arguments, and Technical Handles
argument for an alternative generalization. I can also use it to facilitate discussion about
writing about pedagogy in prose outside of the argument structure.
5.2 Summary of importance
By identifying three technical handles that appear to be important in articulating viable
arguments, instructional interventions are suggested for future studies. Preservice teachers might
benefit from instructional practices that emphasize these three technical handles as skills and
useful moves in developing viable arguments. Preservice teachers might benefit from explicit
instruction that distinquishes between claims that are appropriate based on data and CIs from
those that do not follow prudently from the data and conceptual insights. Explicit instruciton that
specifies types of representations of data and CIs that lend themselves to viable argument
construction for generalizations could benefit learners who might otherwise not have access to
these handles. Explicit instruction on adequate links between data or CIs and claims, particularly
instruction that distinquish between generic and empirical warranting, could benefit learners as
well.
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