Implementing the Third Mathematical Practice by using Argumentation
Implementing the Third Mathematical Practice by using Argumentation
Guanhua Ren
Vanderbilt University
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Abstract
This paper examines the current research and theories that can help those teaching the
third Mathematical Practice in the Common Core State Standards of Mathematics. Driven by
a Constructivist approach to learning, the two existing design models of Problem-Based
Learning and the Hypothetical Learning Trajectory could be helpful for teaching this
Mathematical Practice. In addition, the idea of developing argumentation as a way to promote
student learning becomes most suitable for this kind of lesson content.
However, this poses a challenge to teachers to consider multiple aspects of the
classroom during teaching. When using the concepts of argumentation research, focusing on
class norms could help teachers keep the classroom running smoothly.
Other research on the structure of argumentation provides valuable insight into how to
evaluate argumentation as a learning product. This paper analyzes how Problem-Based
Learning and the Hypothetical Learning Trajectory can help teachers design feasible lesson
content for teaching this Mathematical Practice and the role of the teacher during instruction.
It finally discusses ways to help students develop strong argumentation skills, and
investigates ways to assess argumentation.
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Implementing the third Mathematical Practice by using Argumentation
The Common Core State Standards (CCSS) have been controversial since being released.
In response to the calls for educational reform, CCSS have included many new concepts
within the curriculum. One of these new concepts in the Mathematics Standards is the
introduction of Mathematical Practices. According to the CCSS, these practices are to be
taught in every grade from kindergarten to twelfth grade. However, it is worth noting that
details of how these practices are to be connected to each grade level's mathematics content
are left to local implementation of the Standards. Therefore, it still requires a large amount of
effort to follow through in practice with these Standards.
Mathematical Practices as the Goals in the Common Core Mathematics Curriculum
According to the description in the CCSS, the Standards for Mathematical Practices
describe groups of knowledge that educators ought to develop in their students. By engaging
in learning them, students should be able to develop mathematical maturity and expertise
throughout the elementary, middle and high school years.These practices, which rest on
important “processes and proficiencies” (CCSS, 2010), are based on the National Council of
Teachers of Mathematics (NCTM) process standards of: problem solving; reasoning and
proof; communication; representation; connections and the strands of mathematical
proficiency specified in the National Research Council’s report, Adding It Up, (2011). These
strands are adaptive reasoning; strategic competence; conceptual understanding; procedural
fluency and productive disposition.
After students master these “processes and proficiencies”, they should become fluent in
carrying out mathematical processes. Moreover, such fluency is not acquired by repetition of
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procedures, but by understanding of mathematical ideas. This is a substantive answer to the
problem of “a mile wide and an inch deep”, (CCSS, 2010)
In this article, I am going to focus on the third Mathematical Practice: “Construct viable
arguments and critique the reasoning of others”, and show how students can achieve both
procedural fluency and deep mathematical understanding when they learn this practice.
The third Mathematical Practice requires students to “make conjectures …… analyze
situations …… justify their conclusions, communicate them to others, and respond to the
arguments of others” (p. 6). It is not difficult to see that this practice provides rich ground for
communicating with their peers. In addition, since students are required to make conjectures,
analyze situations and justify conclusions, they will be forced to use their reasoning skills and
prove their stance. Therefore, it is also going to cultivate reasoning and proofing skills for
students, which is another important process in the NCTM standards.
However, as also mentioned in the National Research Council’s report, Adding It Up
(2011), the five strands (adaptive reasoning, strategic competence, conceptual understanding,
procedural fluency, and productive disposition) are “interwoven and interdependent in the
development of proficiency in mathematics”, (p. 116). This means that students will not be
given enough support for their mathematical learning if teachers merely attempt to address
one or two of these strands. Therefore, it is the teachers’ challenge to bring all “processes and
proficiencies” to the classroom while they teach this practice.
Problem Based Learning as a Vehicle for Teaching Mathematical Practice
Problem Based Learning (PBL) is an instructional method in which students learn
through facilitated problem solving. In PBL, students are presented with an ill-defined
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problem that does not have a simple correct answer. Also, students work in collaborative
groups to solve the problem. A PBL approach should be organized around the investigation,
explanation, and resolution of meaningful problems, (Torp & Sage, 2002). However, as
Lampert (2001), points out: “what is to be taught as students work on such problems - and
more importantly how it is to be taught – usually remains vaguely articulated”, (p. 3).
In order to make clear what should be introduced in a PBL classroom and how it should
be taught, Hmelo-Silver (2004) has developed a mechanism called the PBL cycle. In a PBL
cycle, the students are given a problem scenario. They analyze the problem by identifying the
relevant facts from the scenario. Then, they generate hypotheses about possible solutions. An
important part of this cycle is identifying knowledge deficiencies relative to the problem.
Students will follow a process of Self-Directed Learning (SDL) and learn through their
knowledge deficiencies. Then, students apply their new knowledge and evaluate their
hypotheses in light of what they have learned. And lastly, students reflect on the abstract
knowledge gained.
The PBL cycle described by Hmelo-Sliver is a good protocol for teaching mathematical
practice. While students identify relevant facts and generate hypotheses, they will have
opportunities to use their conceptual understanding and reasoning skills. Moreover, because
an ill-defined problem does not have a unique answer or solution path, students will generate
different hypotheses which depend on their strategic competence and procedural fluency.
Then, teachers can create more opportunities for learning because of the diversity among the
students, as I will explore later in the paper.
In the PBL approach, it is critical for the teacher to find a fruitful problem such that the
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students can engage with and learn. In 1997, Margetson suggested that it is necessary to think
about the nature of the problem used to trigger student learning. He argued that within the
PBL context, problems are invariably identified as ‘ill-defined’ and ‘real-world’, pointing out
that they are artificial abstractions specifically constructed to facilitate student learning.
Therefore, it poses the challenge to teachers and educators that they should not only know
how to facilitate PBL, but also understand how to select or create the appropriate problems
with precision.
Research articles about the PBL approach have an almost consistent view about the
characteristics of a good problem. The problem, as Margetson (1997), Torp et al. (2002) and
Hmelo-Silver (2004), all mention, should be “complex”, “ill-structured”, and “open-ended”.
In addition, the problem should provide opportunities for conjecture and argumentation in
order to promote the PBL cycle. The solutions for the problem should be complex as well.
They are often inter-disciplinary or require knowledge from several areas of one discipline.
Last but not least, as Mauffette et al. (2004), point out, although PBL is taken to be inherently
motivating, few researchers have considered the motivating factors for PBL. Yet motivation
is a critical part of teaching. Therefore, Mauffette et al. suggest that educators should also
take motivation into account when they are evaluating the problems. They provide criteria for
motivational problems which include educational goals, background information, setting,
problem, content, resources, and presentation (see appendix A). Following on from these
requirements, I have generated the task below as an example for the PBL approach:
Adam and Bill are two athletes that are preparing for the Olympics. Unfortunately, only
one of them can be offered the place. Given the information below, detailing their sprint times,
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if you are going to recommend one of them, who will you choose?
(The results of the Olympics will also be given to the teacher)
The data is Adam’s and Bill’s results for 9 rounds. All data are measured in seconds.
Round
1
2
3
4
5
6
7
8
9
Adam
19.34
19.71
19.58
20.14
19.58
20.13
19.99
22.62
20.07
Bill
19.97
19.96
20.39
20.13
20.21
20.44
20.18
20.61
19.94
This problem aims to provoke arguments about mean and standard deviation. As the
table shows, Adam has a small mean but large standard deviation whereas as Bill has a larger
mean and smaller standard deviation. This means that Adam is generally faster, but not
always in shape. Bill, on the other hand, is slightly slower, but more consistent than Adam.
The answer to this question, of course, could be either competitor. It is an open-ended
question. In fact, there is not a unique correct answer to it, nor should there be. What is
important here is that this question can trigger students to make conjectures and arguments,
and to justify them. As they work though this question, it will be necessary for them to use
their knowledge, not merely recall it, but also to actively apply it. By doing so, the knowledge
can be refined and consolidated.
The Constructivist Approach as a Mechanism of Learning
Constructivism is a way to describe how knowledge has been built up as a learner
receives and processes information. An essential difference between constructivism and its
previous learning theories (such as Behaviorism, Didactic Theories) is that Constructivism
highlights the importance of prior knowledge which is unique to each learner. As
Fenstermacther and Soltis (2009) claim: “…the teacher is not who imparts knowledge and
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skill to another, but one who helps another gain his or her own knowledge and skills”, (p. 31).
Among all theories of Constructivism, Jean Piaget’s (1953), notions of assimilation,
accommodation and equilibration stand above all others in influence and importance.
According to Piaget, children come into the world with only a few behaviors to guide their
actions. Yet they are actively observing and acquiring new information. During this process,
the children begin to build their initial “schema”, which are clusters of knowledge about a
concept or object (Martinez, 2010).
Although a schema could be a disordered one, children will tend to organize these
clusters of knowledge so that they become a cohesive whole, a “cognitive structure”. This
process, which depends greatly on the knowledge they already have, could possibly produce
a naïve cognitive structure in reality, but sensible to the child. In other words, children are
always willing to reach equilibration in their cognitive structure.
As children keep observing new information, they will first try to make sense of the
information by their previous cognitive structure, their prior knowledge. Then, the new
information and the new explanation will gradually integrate with the existing cognitive
structure. This is the how assimilation occurs. However, since the cognitive structure could be
wrong, it is inevitable that children receive some information that cannot be explained by
their cognitive structure. In these cases, small adjustments to their cognitive structures are not
sufficient to make sense of the new information. Therefore, children have to reconstruct their
cognitive structure. This is the process of accommodation, (Phillips & Soltis, 2009).
In reality, it might take a long time for children to go through the process from the
recognition of cognitive flaws (or in terms of the PBL cycle, the knowledge deficit), to
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reconstructing an advanced cognitive structure. It demands high cognitive load activities,
such as thinking, reasoning and justifying. In Piaget’s theory, all of these activities will take
place inside the child’s head, and are personal rather than social activities.
Nevertheless, there are widely accepted criticisms (Martinez, 2010) about the lack of the
social aspects of learning in Piaget’s theory. It is possible, I believe, to make social activities
into opportunities for learning through class discussion. The Constructivist mechanism is a
valuable approach for teaching Mathematical Practices by employing peer discussion.
In addition to Piaget’s original theories of Constructivism, many scholars have made
further progress with his theory. For example, Simon and Tzur (2012), articulate a
mechanism for mathematics concept development grounded in the construct of assimilation, a
key aspect of Constructivism. This articulation describes how learners’ goal-directed
activities can lead to the generation of more sophisticated concepts. The process begins with
the learners setting goals. These goals are often related to the task posed by the teacher, but
not necessarily the same as the goals that the teacher has in mind. In fact, these two are
different most of the time, and it is critical for the teacher to be aware of such differences.
As learners engage in their activities, they are going to create mental records. Each
mental record is an iteration of the activity linked to its effect, which is similar to the concept
of raw information or observation in the constructivist approach. Then, the learners sort and
compare, which leads to identifying the relationships between the activity and its effects.
These components: creating records of experience; sorting and comparing records, and
identifying patterns in those records, are altogether referred to as a process called “reflective
abstraction” by Simon and Tzur which helps the learners to construct new concepts and new
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activity-effect relationships.
The Hypothetical Learning Trajectory (HLT) as a Tool for Generating Activities
In Simon’s oft-cited 1995 paper, he uses the idea of a learning trajectory as a description
of key aspects of planning mathematics lessons. According to the paper, there are two
important features of a learning trajectory. Firstly, a learning trajectory does not exist in the
absence of an agent and a purpose. And secondly, it is a teaching construct – something a
teacher conjectures as a way to make sense of where students are and where the teacher might
take them. Also, it is hypothetical, because an “…actual learning trajectory is not knowable in
advance.” (Simon M. , 1995)
Since a learning trajectory must have a purpose, this is the first thing that should be
considered. In this article, the purpose of our learning trajectory would be to teach the
Mathematical Practice: “Construct viable arguments and critique the reasoning of others”.
The agent of the learning trajectory is undoubtedly the teacher, and their mission is to connect
the students’ current thinking activity with a possible future thinking activity. Therefore, it is
time for teachers to consider how learners learn and how to bring them to the next level.
In this article, I assume a Constructivist approach. This means that the HLT I must
design will address prior knowledge first in order to find out where students are, and then
give students opportunities to abstract and represent the situation before the teacher explains
the principles.
Following the discussion above, I am now going to generate an HLT for the problem I
posed before. This generation of an HLT begins with identifying a learning goal for students.
The learning goal here will be to learn the statistical ideas: measure of center and measure of
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spread. More specifically, students are going to understand:
1) How to compute measure of center and measure of spread.
2) What measures of center and measures of spread represent respectively.
3) What a large (or small) measure of center and/or measure of spread tells us.
Once the learning goals have been identified, the creation of an HLT involves generating
a hypothetical learning process in the context of a particular set of learning tasks. Using the
mechanism of reflective abstraction, the activity-effect relationship offers a starting point. We
should design an activity in order to begin the process of assimilation. In addition, this task
should be immediately available for the students to do without much effort.
Thus, the first task for the students is to “represent the data set by statistical
measure(s)/tool(s).” This task purposely gives a vague description of the statistical tool that
they should use. While doing so, I expect students to choose the first statistical tools available
to them and therefore let them make their first conjecture. They could use mean, median,
mode, standard deviation, histogram, stem and leaf diagram, pie chart. The idea they choose
should be the most familiar statistical tool for the student. In this way, this first task now
becomes the basis for the intended learning. In addition, the fact that there is no one correct
selection of these tools leaves space for students to justify their selection.
Once students create mental records for this activity-effect relationship during the
learning process, the next task should be designed around the reflection process of mental
records. Thus, the next task of the learning process is to reflect on what they have done, and
comment on their products. The aim that serves for this purpose could be: “Describe what
representation you have used, what information your representation shows, and what it hides
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or overlooks.” While students engage in these two tasks, they will create and reflect on the
effect that their activity has caused, and thus they can assimilate new effect with the activity.
This shows the complete cycle of generating activities and tasks. Then, the teacher can
start again by setting a new goal that can be taught by this problem, then design a new
activity as a basis, and finally provide the opportunities for reflection.
Promoting Argumentation as a Focus during Teaching
The problem and tasks I have developed above address most of the “processes and
proficiencies”. However; I have not yet discussed how this task can be used to cover the
processes of communication, representation, or connection. As a result, the problem does not
have a strong affordance to teach the Mathematical Practice: “Construct viable arguments and
critique the reasoning of others”. In order to enhance this affordance, it is necessary to
consider the role of teacher during this learning process.
In my previous discussion, the teacher serves as the agent in HLT to identify the goals,
and to promote the establishment of the activity-effect relationship as well as the reflection of
it. The teacher is also the facilitator in PBL in order to guide students to find the knowledge
deficits and resources for students to process though SDL (Self-Directed learning).
Now, I propose that it is also the teachers’ responsibility to promote argumentation and
class discussion as the products of this activity. When teachers fulfill this role, the processes
of communication, representation, and connections should all be covered and thus students
will have a better opportunity to learn Mathematical Practice.
In fact, discourse in classrooms has always been viewed as a key aspect in teaching and
learning. Margaret and Glenda (2008), refer to verbal discourse as “…acceptable, indeed
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essential, in the classroom, and mathematical discussion, explanation, and defense of ideas
become defining features of a quality mathematical experience.” (p. 516). In addition,
teachers should invite students to: “…develop explanations, make predictions, debate
alternative approaches to problems. . . (and) clarify or justify their assertions”, (Brophy, 2001,
p. 13) in order to develop an understanding of the big ideas and an “appreciation of its value,
and the capability and disposition to apply it in their lives outside of school”, (Brophy, 1999,
p. 4). In concurring with Brophy’s stance, Yackel and Cobb (1996), find that students become
less engaged in solutions to problems, than in the reasoning and thinking that lead to these
solutions.
However, it is worth clarifying that this kind of argumentation is not what usually occurs
on talk shows or in the political sphere! The argumentation in classrooms is a form of
collaborative discussion in which both sides share the same assumptions and work toward the
same goal. When commenting on the concerns about argumentation in the classroom,
Walshaw and Anthony (2008), even claim that students cannot learn mathematics with
understanding, without engaging in discussion and argumentation.
The Teacher’s Role during Instruction
Although teachers should bring argumentation into the classroom, it does not imply that
more discussion in the classroom leads to better understanding. In order to promote high
quality argumentation, teachers have to interact with students wisely. Stein et al. (2008)
suggests four practices that teachers should attempt during teaching. These practices are
monitoring, selecting, sequencing, and connecting.
According the Stein et al., a teacher should be monitoring while students are working on
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a task. While citing Stein et al.’s idea, Horn (2012) elaborates that a teacher should “circulate
(s) around the room, listening but not hovering. The teacher pays attention to student thinking
and group dynamics, intervening only occasionally”, (p. 67). While monitoring, the teacher
has to pay attention to both the mathematical and the social aspects of class work. This means
that, on the one hand, the teacher should listen to and recognize the kinds of mathematical
thinking occurring during the working period. On the other hand, they should step in
whenever the group has stopped trying, or is not engaging. Teachers are required to have
knowledge in two areas, as Sherin (2002) states: the subject matter knowledge and the
pedagogical content knowledge.
Then, the teacher has to select what should be presented to the whole class. It could be
an unexpected solution(s), or it could be a disagreement which is raised during the working
period. It is also very possible that there are many solutions or questions that the teacher
wants to discuss with the whole class. This means that the teacher has to sequence these
topics into a purposeful order. For instance, the teacher certainly does not want a student to
present a neat answer while there is still disagreement or confusion. Instead, the teacher can
ask students to come and present their argument on such a disagreement, or their confusion,
and ask the rest of class to comment. If a teacher has conducted the lesson in this way, the
lesson becomes exactly what is emphasized in the Mathematical Practice: “Construct viable
arguments and critique the reasoning of others”.
Finally, novice students might not be able to fully understand others’ reasoning, or the
connection between these differently presented arguments. Therefore, it is the teacher’s job to
make the connection clear to them. When the connection has been established, the students
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can make sense of different solutions, connect to their original thinking, and thus construct
new knowledge by promoting assimilation or accommodation.
Besides the mathematical aspects, a teacher should also be monitoring other aspects of
the classroom. Allen and Blythe (2004), identify several challenges that a teacher could be
facing. These challenges include: running out of time; having a big group; students easily
going off-task; students being reluctant to offer comments, and many others. In order to deal
with these issues, Allen and Blythe suggest that the teacher should be planning ahead with the
anticipated problems in mind, and try to give every student the chance to participate. Similar
suggestions also occur in Horn’s work (2012).
Norms as a guideline for student behavior during discussion
In many research articles, scholars have identified norms within the classroom
community as a crucial for this purpose. One reason is that, as students first participate in a
discussion, they often struggle with when and how to contribute to mathematical discussions
and what to do as a listener. Setting up appropriate norms can help students learn. In addition,
developing helpful norms in the classroom can provide more time for teaching, rather than
classroom management. However, some norms can be implicit and hinder students’ learning.
As Bransford (2000) proposes, norms like: “Never to get caught making a mistake or not
knowing an answer”, can hinder a student’s willingness to ask questions. Helpful norms that
aid classroom management are referred to as general classroom social norms. Examples of
this type of norm could be: raising a hand before speaking, or, not talking while others speak.
To be specific, there is one strategic norm that has been mentioned in many studies. It is
to assign group roles to students during discussion. Horn (2012), offers four roles as follows:
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facilitator, resource monitor, recorder/reporter and team captain. By assigning these roles, the
students know how they are supposed to behave, and thus help the class to follow norms.
To extend the idea of norms, Yackel and Cobb (1996), introduce the notion of
socio-mathematical norms, to be distinguished from general classroom social norms.
According to Yackel and Cobb, socio-mathematical norms are norms that focus on normative
aspects of mathematics discussions, specific to the students’ ongoing mathematical activity.
Their examples of socio-mathematical norms are: normative understandings of what counts
as mathematically different, mathematically sophisticated, mathematically efficient and
mathematically elegant in a classroom. An example of socio-mathematical norms can be, as
discussed in the problem above, that students are expected to find an agreement by the end of
their discussion instead of a disagreement.
Unlike general social norms, engaging socio-mathematical norms can directly initiate
learning. Also, since learning rarely happens by direct instruction, the implementation of
socio-mathematical norms takes a lot more than just repetition. For example, when two
students pose two different solutions or arguments, the teacher can ask the class why they are
different. Then, students will justify their solutions as well as comment on others. This is the
necessary process of implementing socio-mathematical norms. At the same time, as students
engage in this process, they can also experience higher-level cognitive activity. Therefore,
Mathematical Practice can also be taught during this process, since it provides many
opportunities for them to communicate and justify.
Evaluation of Students’ Argumentation
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According to Andriessen (2006), the study of argumentation has been dominated for
most of the last century by scholars who focused on the sequential structure of an argument.
This implies that a good argument should have a certain type of structure, which includes a
claim, data, warrant, backing, qualifier, and a rebuttal if there is an exception (Toulmin, 1958).
To take the problem I have designed above as an example, a student may say that he
recommends Adam to participate in the Olympics. This is his claim for this problem. Then,
either by knowing the norm that everyone should provide warrants for their claim, or by
questioning from his peers or teacher, he should be aware that there are some aspects missing
from his argument. Therefore, the student may continue to construct his argument. He may
say that he chooses Adam because Adam has a smaller mean. Then, a more subtle part of the
argument is required, the backing. The student should also explain why mean is a critical
measure but not variance, median, mode, or other measures. Finally, the student can give his
conclusion with a suitable qualifier and rebuttals if necessary. Cobb (2002), and Berland and
McNeill (2010), also apply Toulmin’s scheme of argumentation in their research framework.
Cobb provides a graphical structure for Toulmin’s scheme (Appendix B).
However, let us now consider the scenario in which another student comes and argues
that Bill should be offered the place. How will the first student respond? Toulmin’s scheme
does not provide a structure for this situation. Both Andriessen and Cobb recognize this
shortcoming in Toulmin’s work. Cobb acknowledges that Toulmin's scheme has frequently
been criticized, not least because of its Structuralist orientation. In addition, Andriessen
(2006), claims that “the model (Toulmin’s scheme) fails to consider both sides involved in
(real-world) argumentation; it covers only the proponent, not the opponent,” (p. 3)
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To redress this deficiency, Andriessen suggests the Dialogue Theory, which has been
developed by Walton (2000). Dialogue Theory views the interaction between proponent and
opponent as a move made in a dialogue in which two parties attempt to reason together. Six
types of dialogue are described: persuasion, inquiry, negotiation, information-seeking,
deliberation, and eristic (personal conflict), to be used as “a normative model to provide the
standards for how a given argument should be used collaboratively.”
Using Walton’s Dialogue Theory, the interaction between these two students may look
like this: the second student (John) suggests that variance maybe more important for this
situation – the persuasion part. Then the first student (David) will ask John why his argument
should be accepted – the inquiry part. John may say that a better athletic should be more
consistent, and the difference between their means is not that significant. David may disagree,
and they (as well as the rest of the class) will argue back and forth – the negotiation part.
Then they might decide to find out the results of the actual Olympic athletics – the
information-seeking part. Finally, while they have the new information and rethink this
problem, it is hoped that they will come to an agreement – the deliberation part.
In fact, as I mentioned earlier in this article, it does not matter which person they have
chosen. There is no correct answer for this problem. The learning occurs when they engage in
the process of convincing others or are convinced by others. Thus, a teacher who is going to
assess their learning should look for the warrant and backing parts of their argument, and the
persuasion, inquiry, negotiation, information-seeking, deliberation parts of their conversation.
Once these aspects have all been presented, and they are logically linked, the teacher can
conclude that the students have made their progress through argumentation.
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Concerns about the Learner
Besides Constructivism, Piaget is also well-known for his theory of cognitive
development. His intellectual development theory of developmental stages claims that any
learner would experience four developmental stages as they develop from newborn baby into
adulthood. They would begin with the Sensorimotor stage, in which they can only experience
the world through movement and senses. Generally after the age of 2, they will process to the
next stage called the Preoperational stage. This stage includes an increase in language ability
(with over-generalizations), symbolic thought, egocentric perspective, and limited logic,
(Ojose, 2008). The next stage, the Concrete Operational Stage, starts from ages 7 to 11. In
this stage, learners begin to think logically, but are very concrete in their thinking. They can
converse and think logically but only with practical aids. Also, from this stage, they are no
longer egocentric. The final stage is the Formal Operational Stage. In this stage, learners
develop abstract thought and can easily converse and think logically.
Bearing Piaget’s theory in mind, educators should ensure that the problems and activities
they have designed for learning should be modified depending on the developmental level of
the learners. For example, if the learners are still in the concrete operational stage, it is not
sensible for the educators to introduce the highly abstract thinking of advanced ideas such as
groups or infinity, especially when detailed scaffolding is not provided. The learners will not
be ready to understand them; and if they do encounter something like this it will confuse
them completely. It is possible for them to enter a state of learned helplessness if they keep
confronting knowledge that is far beyond their level, (Martinez, 2010). For the learner
experiencing learned helplessness, there is a complete disconnection between personal
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initiative and desired outcomes. Then, the strategy that makes the most sense is simply to
stop trying.
However, even though older learners are potentially more capable of making logical
arguments, it is not always the case. In Berland and McNeill’s (2010), study, they found that
younger learners may perform better than older learners. This could possibly be a result of the
students’ previous experience of argumentation. Indeed, even if the learners are
developmentally ready for learning argumentation, there is a natural tendency to be in favor
of their own side, but not the opponent’s. As a result, they will tend to consider their
argument as a more legitimate stance even if both sides’ arguments are equally reasonable.
The appreciation of an opponent’s position comes a lot later. Therefore, for teaching practice,
a teacher should postpone the debating aspect of argumentation but focus on helping students
articulate their own arguments.
However, this does not mean that students cannot criticize each other. They can talk
about the structural completeness of others’ arguments. This can be viewed as
peer-assessment. When students have developed the skill of making a good argument, then
will be the time to ask them to focus on criticizing the content of others’ arguments as well as
the structure.
Questions and Implications
Forty-five states have adopted the CCSS for Mathematics up until January 15 2013. This
means the curricula of most states will be based on the CCSS for Mathematics by 2015.
Mathematical Practices is a critical part of this curriculum. As stated in the Standards, the
implementation of this curriculum should always: “attend to the need to connect the
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mathematical practices to mathematical content in mathematics instruction”, (CCSS p. 8).
However, the details about implementing these practices have only vaguely been stated. In
addition, over time, the research on learning progressions will inform and improve the design
of standards - to a much greater extent than is possible today. Therefore, continued research
and investigation is essential for successfully teaching and broadening these standards in the
future.
I am aware that this analysis about teaching Mathematical Practice relies heavily on the
theory of Constructivism and the works of Piaget. His work, which is generally referred as
cognitive constructivism, focuses on a sequential development of mental processes. However,
I believe students can learn by interaction with others. Therefore, a possible direction for the
paper is to consider the social constructivism theory by Vygotsky (1978), which claims
experiential learning occurs through real life experience to construct knowledge. Situated
learning (Lave & Wenger, 1991), on the other hand, is another direction for further study. It
discusses how learners explore available tools and learn within a community. It draws on the
analysis and use of tools that may enhance student learning, and is to be considered.
Finally, there are not any two learners who learn in the same way. Therefore, educators
should always focus on answering the question: ‘How can teaching practice best enhance all
students’ learning?’ The introduction of Mathematical Practices is a move of merit, but it also
leaves many questions for educators to consider. If we can make progress in answering these
questions, our students will unquestionably be the beneficiaries.
Conclusion
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The Common Core State Standards delivers a new challenge for teachers. One challenge
is to successfully implement Mathematical Practices into classroom teaching practice. In
order to address this issue, I have analyzed the Common Core Mathematics Standards and
suggest that argumentation could be a focus for teaching. After reviewing research on
teaching and argumentation, I draw three conclusions as follows:
1) In order to teach argumentation, we first have to create appropriate learning contexts.
I suggest the framework of Problem-Based Learning. Then by using the idea of the
Hypothetical Learning Trajectory, teachers should construct complex task sequences,
incorporating many argumentation activities over an extended period.
2) The learning environment is crucial for teaching argumentation. In order to provide a
good learning environment, teachers should establish healthy norms that help the classroom
run smoothly. Moreover, establishing socio-mathematical norms are an essential part of
students’ learning.
3) Research on argumentation suggests a structural feature of argument. Teachers can
assess students’ progress by identifying the quality of different parts of their argument, and by
looking at the completeness of their argument.
Although teachers are no longer the authority in the classroom while teaching
argumentation, this does not imply teachers have a lesser load. In fact, teaching
argumentation required a long term effort in developing a consensus about what counts as a
good argument, what counts as mathematically justified, and what constitutes
socio-mathematical norms. Although these require much more action then just instructing, it
is an efficient way to build a mathematical mindset in students.
Implementing the Third Mathematical Practice by using Argumentation
Appendix A
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Implementing the Third Mathematical Practice by using Argumentation
Appendix B
Graphical representation of Toulmin’s scheme of argumentation
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Implementing the Third Mathematical Practice by using Argumentation
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Bibliography
Hmelo-Silver, C. E. (2004). Problem-Based Learning: What and How Do Students Learn?
Educational Psychology Review, Vol. 16, No. 3, 235 - 266.
Allen, D., & Blythe, T. (2004). The Facilitator’s Book of Questions. New York: Teachers
College Press.
Andriessen, J. (2006). Arguing to Learn. In K. R. Sawyer, The Cambridge Handbook of the
Learning Sciences (pp. 443-459). Cambridge: Cambridge University press.
Berland, L. K., & McNeill, K. L. (2010). A Learning Progression for Scientific
Argumentation: Understanding Student Work and Designing Supportive Instructional
Contexts. Science Education, 94(5), 765 - 793.
Bransford, J. D., Brown, A. L., & Cocking, R. R. (2000). How People Lean: Brain, Mind,
Experience, and School. Washington, D: National Academy Press.
Brophy, J. (1999). Teaching (Educational Practices Series No. 1). Geneva, Switzerland:
International Bureau of Education.
Brophy, J. (2001). Introduction. In J. Brophy, Subject-specific instructional methods and
activities (pp. 1 - 23). Amsterdam: JAI.
Case, R. (1996). Changing Views of Knowledge and Their Impart on Educational Research
and Practice. In D. Olson, & N. Torrance, The Handbook of Education and Human
Development: New Models of Learning, Teaching and Schooling (pp. 75 - 99).
Malden, MA: Blackwell Publishers.
CCSS. (2010). Standards for Mathematical Practice. Retrieved from Common Core State
Standards
Initiative:
Implementing the Third Mathematical Practice by using Argumentation
Ren
26
http://www.corestandards.org/Math/Content/introduction/how-to-read-the-grade-level
-standards
Cobb, P. (2002). Reasoning with Tools and Inscriptions. The Journal of The Learning
Sciences, 11(2&3), 187 - 215.
Fenstermacher, G., & Soltis, J. (2009). Approaches to Teaching . New York, NY: Teachers
College Press.
Horn, I. S. (2012). Designing for Group Work: the Teacher's Role Before, During, and After
Instruction. In I. S. Horn, Strength in Numbers: Collaborative learning in Secondary
Mathematics (pp. 61 - 76). Reston, VA: NCTM.
Kilpatrick, J., Swafford, J., & Findell, B. (2001). Chapter 4: The Strands of Mathematical
Proficiency. In N. R. Council, Adding it up: Helping children learn mathematics (pp.
115 - 155). Washington, DC: National Academy Press.
Lampert, M. (2001). Teaching problems and the problems of teaching. New Haven: Yale
University Press.
Lave, J., & Wenger, E. (1991). Situated Learning: Legitimate peripheral participation.
Cambridge: University of Cambridge Press.
Margetson, D. (1997). wholeness and educative learning: the question of problems in
changing to problem-based learning. Brunel University, UK: Keynote Speech
Presented to Changing to Problem-Based Learning Conference.
Martinez, M. E. (2010). Learning and cognition: the design of the mind. Columbus, Ohio:
Merrill.
Mauffette, Y., Kandlbinder, P., & Soucisse, A. (2004). The problem in problem-based
Implementing the Third Mathematical Practice by using Argumentation
Ren
27
learning is the problems: But do they motivate students? In Y. Mauffette, P.
Kandlbinder, & A. Soucisse, Challenging research in problem based learning (pp.
11-25). Open University Press.
Ojose, B. (2008). Applying Piaget’s Theory of Cognitive Development to Mathematics
Instruction. The Mathematics Educator, Vol. 18, No. 1, 26 - 30.
Phillips, D. C., & Soltis, J. F. (2009). Perspectives on Learning. New York: Teahcer College
Press.
Piaget, J. (1953). Logic and Psychology. Manchester: Manchester University Press.
Shaughnessy, M. J., Barrett, G., Billstein, R., Kranendonk, H. A., & Peck, R. (2004).
Navigating through Probability in Grade 9-12. Reston VA: National Council of
Teachers of Mathematics.
Sherin, M. G. (2002). When Teaching Becomes Learning. Cognition and Instruction, Vol. 20,
No. 2, 119 - 150.
Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective.
Journal for Research in Mathematics Education, 26(2), 114 - 145.
Simon, M. A., & Tzur, R. (2004). Explicating the Role of Mathematical Tasks in Conceptual
Learning: An Elaboration of the Hypothetical Learning Trajectory. MATHEMATICAL
THINKING AND LEARNING, 6(2), 91 - 104.
Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating Productive
Mathematical Discussions: Five Practices for Helping Teachers Move Beyond Show
and Tell. Mathematical Thinking and Learning, 313-340.
Torp, L., & Sage, S. (2002). Problems as Possibilities: Problem-Based Learning for K–12
Implementing the Third Mathematical Practice by using Argumentation
Ren
28
Education, 2nd edn. Alexandria, VA.: ASCD.
Toulmin, S. E. (1958). The Uses of Argument. Cambridge, UK: University Press.
Vygotsky, L. (1978). Mind in Society. London: Harvard University Press.
Walshaw, M., & Anthony, G. (2008). The Teacher’s Role in Classroom Discourse: A Review
of Recent Research Into Mathematics Classrooms. Review of Educational Research
Vol. 78, No. 3, 516 - 551.
Walton, D. (2000). The place of dialogue theory in logic, computer science and
communication studies. Synthese Vol 123, 327-346.
Yackel, E., & Cobb, P. (1996). Sociomathematical Norms, Argumentation, and Autonomy in
Mathematics. Journal for Research in Mathematics Education Vol. 27, No. 4,, 458 477.