MATHEMATIZING CONSTRUCTIVISM 1

Mathematizing Constructivism:

An exploration of percentages, decimals and fractions in relation to area in grade 7 & 8

mathematics.

Sarah Irwin­Gibson & Laura Hall

Dr. Samia Khan

ETEC 530 /65A

March 31, 2016

University of British Columbia

MATHEMATIZING CONSTRUCTIVISM 2

INTRODUCTION

Mathematics is the study of “Principles relating to a specified phenomenon, [or] process”

(“mathematics, n”). This is critical in the development of students understanding of concepts and

creation of their own knowledge. In the revamped BC curriculum, mathematics is described as

holding an integral role in our everyday lives, explaining, “Mathematical values and habits of

mind go beyond numbers and symbols; they help us to connect, create, communicate, visualize,

reason, and solve” (Building Student Success ­ BC's New Curriculum. n.d.). In the 1996

manifesto, the New London Group, describes the mission of education as having the,

“Fundamental purpose [...] to ensure that all students benefit from learning in ways that allow

them to participate fully in public, community, and economic life” (p. 60). Daily life problems

are increasingly emphasized in recent mathematics curricula in various countries around the

world (Singer and Moscovici, 2008, p. 1616). Exploring reality, moves learning from

teacher­driven, to research­driven, where the student is given the power to personalize.

KNOWLEDGE & REALITY

Why do we study mathematics?

Reality plays an instrumental role in the way that we construct knowledge and understand the

world around us. Pritchard (2014) explains, “The way the world appears and the way that it

really is could be drastically different” (p. 73). This, in turn, helps us in environments of inquiry

to construct and question what we know. Voithofer (2005) observes, “We change the world by

changing the way we make it visible,” drawing our attention to the fact that it is not just the

material being created but also the way in which the material is presented and accessed (p. 3).

MATHEMATIZING CONSTRUCTIVISM 3

Mathematics offers a unique approach to visualizing and creating meaning from the world

around us. Mathematics challenges us to view the world in a different way through the use of

numbers and concepts.

Constructivism centers on “The constructed nature of the various realities we inhabit, the

fields we study, [and] the discourses in which we engage” (Greene, 2005, p. 121­122). How we

build our understanding impacts what we know and how we come to know it (Pritchard, 2014).

Constructivist pedagogies support the building of knowledge through connections between prior

and new knowledge, providing a basis for establishing personal and social understanding

(Doolittle & Hicks, 2001). O’Donnell (2006) expresses an importance of communities of

collaboration in the development of student understanding within constructivist principles in

stating, “Such collaborations have the potential to increase the quality of discourse, provide

alternative explanations for phenomenon, generate multiple solutions to problems, and allow for

the inclusion of many different kinds of skills in problem solving” (p.4). From Galileo to

Einstein, the world is effectively constructed to support creation of connections through

mathematics and our reality.

The instruction of mathematics has traditionally been transmitted through the delivery of

instructions by teachers. As Fosnot (2013) outlines, “Most students in mathematics classrooms

[do not] see mathematics as creative but instead as something to be explained by their teacher,

then practiced and applied” (57%). Constructivist pedagogies challenge the traditional delivery

of mathematics to encompass a more holistic approach. Fosnot (2013) states that mathematicians

engage quite differently: “They make meaning in their world by setting up quantifiable and

spatial relationships, by noticing patterns and transformations, by proving them as

MATHEMATIZING CONSTRUCTIVISM 4

generalizations, and by searching for elegant solutions” (58%). If we offer our students

opportunities to engage in the exploration of concepts as mathematicians would, we are offering

them a chance to be creative and build real relationships, as they communicate their learning

with others.

We want to offer our students a chance to investigate grounded within a context, in order

to generate mathematizing moments (Fosnot, 2013, 57%). These moments ask for students to be

immersed in their surroundings finding opportunities to extend their understanding through

connections to mathematical observations. Naylor (2016) observes, “Taking mathematics in

context, and providing students with real world, real life opportunities to utilize math generates a

creative community of practice founded around constructivism” (Naylor, 2016). To meet the

students cognitive needs, we must bridge conceptual knowledge to application of such

knowledge (Pritchard, 2014). The goal is to develop autonomous students capable of engaging in

personally meaningful inquiry (Doolittle P.E. & Hicks, 2001, p. 17). Piaget explored this as the

“Mapping of actions and conceptual operations that had proven reliable in the knowing subject’s

experience” (von Glasersfeld, 2005, p.3). This is crucial as we aim to satisfy students

information­gap, or rather curiosity, through the exploration of topics and how mathematics can

support their understanding and further development in the field (Arnone, et al., 2011).

Supporting students interest helps to foster an active community of learners as they are

intrinsically motivated and engaged with the material as it is developed through zones of

proximal development (Vygotsky, 1978). This inquiry­based approach to instruction supports, as

Singer and Moscovici (2008) explore, an enhancement of quality of student achievement

(p.1633).

MATHEMATIZING CONSTRUCTIVISM 5

By offering concrete opportunities to inquire, we are ensuring that students do not access

knowledge as “Simply a matter of luck” (Pritchard, 2010, p.7). Rather, it creates an authentic

exploration and engagement of experiences. Smith and Smith (2014) note that engaging in

authentic learning situations that include challenging work, immediate feedback, learning

choices and social interactions improve student outcomes. Providing these opportunities values

students experiences in life and learning in their own individual way as it shapes how they come

to understand beyond the confines of their mind (von Glasersfeld, 2005, p. 201).

LEARNING IS SITUATED

How can the study of mathematics offer situated learning experiences for students?

Constructivism shifts our focus from overarching subject areas to inherently placing the

responsibility of knowledge acquisition into the hands of the learner. The inquiry­based

philosophy that lends to hands on experience, requires students to be a part of a participatory

culture as created by the instructor (Siegel, 2012). This is illustrated by Forman (2013) through

his emphasis on collaboration mindset of constructing knowledge as a collective (66%).

Constructivist theory allows for cross­curricular connections. This allows for growth and

development, through situated learning opportunities. Situated learning happens in classrooms

“embedded in the real­life experiences of children, and with rich discourse, about mathematical

ideas” (Small, 2013, p.3). Consistent incorporation of situated learning opportunities offers

students the chance to connect with knowledge and engage in meaning making (Arnone et al.,

2011). Cobb (2005) explains, “the concept that learners actively construct their knowledge as

they interact with the world is interconnected with the concept that learning is socially and

MATHEMATIZING CONSTRUCTIVISM 6

culturally situated” (Chapter 3). Learning does not occur in isolation ­ rather, it is through

interactions and creating a community of learning that supports Vygotsky’s theory of social

interaction (McKenzie, 2000). Situated learning is developed through interconnected

relationships with self, peers, and content. Therefore, students are provided multiple entry points

through facilitated connections and communications that acknowledge and value their

experiences.

ELICITING PRIOR KNOWLEDGE

What is the importance of creating connections with previously held knowledge and skills?

Constructivist pedagogies support student learning through opportunities to create a community

of learners. This blending of education and student curiosity is tapping into a more holistic way

of learning ­ one that caters to the social and academic self, while creating a more personalized

and focused experience. Rennie and Mason (2008) explore this stating that students “Actively

engage in the construction of their experience, rather than passively absorbing existing content”

(p.4­5). In order to promote student engagement and participation, it is necessary that as teachers

we are able to elicit prior knowledge in order to scaffold and prepare meaningful content.

Without providing opportunities to elicit prior knowledge, students lack interest and ability to

connect information to self which is fundamental in order to create meaning (Baviskar, 2009,

Snively & Corsiglia, 2000, Nola, 1997). Preliminary knowledge as explored by Barnes, is

elicited through discussions, mind­maps, visuals, and reflection on connections students create

within the given context (Sunal, n.d.).

As Piaget stated in 1977, “The mind is never a blank slate. Even at birth, infants have

organized patterns of behaviour ­ or schemes, for learning and understanding the world” (Fosnot,

MATHEMATIZING CONSTRUCTIVISM 7

2013, 59%). When we ask students to access prior knowledge in order to seek out patterns that

they see in relation to concepts at hand, we are allowing them to construct and develop new

schemes to explain what they are experiencing. Lutz & Huitt (2003) support this in their

description of knowledge, stating, “[…] Knowledge units are not simply stored and then left

alone, but that they are retained, manipulated, and changed as new knowledge is acquired” (p. 2).

Eliciting prior knowledge is fundamental when examining mathematical concepts in

order to support authentic engagement and understanding on the part of the students. Cobb

(1988) states, “That the two goals of a constructivist approach to mathematics are students’

opportunity to develop richer and deeper cognitive structures related to mathematical ideas, and

students’ development of a level of mathematical autonomy” (Small, 2013, p.4). In order to

support a deeper understanding through mathematical concepts, students must first understand

relationships within their own knowledge.

CREATING COGNITIVE DISSONANCE

How does reality affect our ability to learn and to engage with concepts?

Baviskar et al.’s (2009) second tenet of constructivism “creating cognitive dissonance” is

understood to be the tension between information and ideas (Baviskar et al, 2009, p. 541,

Kennedy, 2016). Merriam Webster describes this to be the “psychological conflict resulting from

incongruous beliefs and attitudes held simultaneously” (“cognitive dissonance, n”). This

juxtaposition of comfort and challenge strengthens students understanding of content area.

Although increasing in interest for many, it is essential that we learn to differentiate for students

to effectively support student development of content. Creating a framework using open­ended

MATHEMATIZING CONSTRUCTIVISM 8

questions to explore, offers an environment where students are active participants ­ compelling

them to engage through their discomfort and take risks, venturing forward as they develop a

personalized understanding. Capitalizing on their experiences from drawing on their repertoire of

mathematical concepts (prior knowledge) and their understanding of reality, students creatively

explore cognitive dissonance as they navigate through collaborative, participatory and individual

exercises (Pritchard, 2014).

Reality affects our ability to learn in a variety of ways. From instrumental value to

experiential knowledge we attempt to come to understand true belief depending on the way we

choose to engage (Pritchard, 2014). Students must first of all “Play an active role in selecting and

defining the activities, which must be both challenging and intrinsically motivating” (Fosnot,

2005, 32%). Eliciting prior knowledge is fundamental in engaging and participating in activities

of cognitive dissonance. Shalni Gulati (2008) describes pedagogy through the eyes of Bruner

(1999) as “A science that involves becoming aware of the different learning strategies and how,

for whom, and when to apply these strategies” (p. 183). The role of the teacher remains to

incorporate these elements through facilitating experiences. Robertson (2008) identifies that it is

essential that students “Internalize [...] the exact fit between the curriculum [and inquiry]” to

support their growth and development (p. 58). This needs to exemplify not only student

engagement but ability to manipulate and create personally relevant meaning within the learning

context (Lamey, 2016). The usage of constructivist pedagogies through a collective community,

students and teachers alike are invited to engage and create meaning to support their practice and

overall understanding of content (Hattie & Timperley, 2007, p.103). Ultimately, we want to

MATHEMATIZING CONSTRUCTIVISM 9

create “The greatest opportunities for students to learn, regardless of the techniques used”

(Baviskar 2009, p. 542) and offer effective situations for mathematization to occur.

REFLECTIVE PRACTICE

How can reflection help us to organize our beliefs?

Reflection is an essential in our construction of knowledge (Moon, 2001). Creating linkages

between existing information and new knowledge, affords us the opportunity to create deep

reflection in a way that supports cognitive development (Moon, 2001, p. 5, Pritchard, 2014,

Fosnot, 2013). Knowledge is not intended to be acquired in isolation, rather it is meant to help

design our thinking as we manipulate, revise and apply it to a variety of new environments (Lutz

& Huitt, 2003, p. 2, Cambridge, n.d., p. 7). Vygotsky’s constructivist theory calls for students’ to

be active participants in their construction of knowledge in order to draw thoughtful and

meaningful information (Schneuwly, 2008). This process allows students to not only recall the

information learned but also to come to understand how it has changed their understanding.

Von Glasersfeld (2008) explains, “To see it and gain satisfaction from it [construction of

knowledge], one must reflect on one’s own constructs and the way in which one has put them

together” (p. 48). Throughout the unit, students are challenged to reflect on their own work as

well as that of their peers. Reflection helps us to organize our beliefs as we create opportunities

to visualize, through personal and peer feedback, and within collaborative environments. Piaget

(1971) believed that the action of reflective abstraction was the “Mechanism by which all

logico­mathematical structures are constructed” (Dubinsky, 1991, p. 160). By encouraging our

MATHEMATIZING CONSTRUCTIVISM 10

students to think about their learning, we are offering them a chance to construct a more

meaningful understanding of what they observe and in turn, know.

As explored in Knowledge and Constructivism (A1, Hall, 2016), von Glasersfeld (2008)

implies that there is less value in truths and facts. Rather, the value lies within the process – to

learn to construct, reflect and evaluate ideas in relation to our surroundings. The non­linear

approach that constructivism allows affords students the opportunities to fully interact, generate

connections and create individualized learning through the participatory culture needed to

engage in civic life (Fosnot, 2013, 5%, von Glasersfeld, 2008, Siegel, 2012, New London Group,

1996). This, in turn, changes the role of the teacher to facilitator of an environment of inquiry

where students are supported through development of skills for questioning and scaffolding to

meet individual students needs (Wu & Huang, 2007). Knowledge cannot be transferred but

rather, must be constructed.

CREATING A COMMUNITY OF PARTICIPATION THROUGH CONSTRUCTIVIST

PRACTICES IN MATHEMATICS

Constructivism requires students to be active participants in the pursuit of information. It

encourages the creation of knowledge through collaboration, information building, reflection and

inquiry. Siegel (2012) states,

Students will need to become designers of meaning with facility in the full range of design elements or modes of meaning making – including visual, audio, gestural, spatial and multimodal meanings – in order to successfully navigate the diversities of texts, practices, and social relations that are part of working lives, public lives and personal lives in ‘new times,’ further extending the need for participation for meaningful and practical learning. (p. 672)

MATHEMATIZING CONSTRUCTIVISM 11

Barnes’ method for the implementation of constructivist strategies proposes four stages

of development: focusing phase ­ preliminary stage where information is presented (1),

exploratory phase ­ where students are engaged in constructing ideas (2), reorganizing phase ­

where students come to make sense of their exploration through creating a chain of ideas (3) and

finally, a public phase ­ presenting occasions for students to share their discoveries (4) (Sunal,

n.d.). This final stage requires students to publicly present their findings with one another, in

order to lead the group towards further discussion and debate (Sunal, n.d.). By offering students

a chance to engage online through webquest explorations, in public forums such as mathematical

congresses and through gallery walks, we provide opportunities for them to learn from each

other and through their shared experiences. Willis (2010) explains that the learning process,

“Start[s] the unit with global connections to students’ lives through discussion; demonstration;

[...] facts; or humor related to the topic” (p. 140). Constructivist pedagogies emphasize the

importance of collaboration in order to support learning as a whole. Dooly (2008) states,

This may include students teaching one another, students teaching the teacher, and of course the teacher teaching the students, too. More importantly, it means that students are responsible for one another’s learning as well as their own and that reaching the goal implies that the students have helped each other to understand and learn. (p. 1)

Constructivist teaching practices help students to gain a better understanding of their

knowledge and to build from previous experiences, all the while expanding their global

understanding of their surroundings (Pritchard, 2014, p. 59). When students are offered a chance

to mathematize their own ideas, through constructivist approaches to education, they “Will come

to see mathematics as the living discipline it is, with themselves a part of a creative, constructive

mathematical community, hard at work” (Fosnot, 2013, 60%). Baviskar et al. (2009) state the

importance of a constructivist lesson as, “one that is designed and implemented in a way that

MATHEMATIZING CONSTRUCTIVISM 12

creates the greatest opportunities for students to learn, regardless of the techniques used” (p.

542).

CONCLUDING REMARKS ON MATHEMATICAL CONCEPTS

Mathematics, and education as a whole, extends beyond the confines of the four walls to which

academic learning occurs. To support student development of concepts, we must create

connections to support real world opportunities that allow mathematics to extend from

theoretical to practical application. Koshy et al. (2009) explains students need to “Acquire an

understanding and appreciation of mathematics and its importance in their lives” (p. 214). This is

reinforced as students are encouraged to perceive knowledge as they see fit in order to support

their learning (Pritchard, 2014, p. 59). This is an important task as students are encouraged to

revisit and construct meaning from the content being facilitated, as philosopher W.V.O. Quine

states, “no statement is immune to revision” (Pritchard, 2014, p. 35). Students are not only

required to master their understanding but come to question and revisit the truths they have come

to understand.

Constructivism is not a method nor a teaching model, it is a philosophy that can

contribute to critiquing and problematising of existing and emerging educational practices

(Larochelle, Bednarz & Garrison, 1998). The ultimate goal of a collaborative learning process, as

outlined in constructivist pedagogies, is to “Assist teaching in a specific educational objective

through a coordinated and shared activity, by means of social interactions among the group

members” (Zurita & Nussbaum, 2004). Mathematization of concepts brings learning to life.

Fosnot (2013) explains that “An engagement, a conversation, a quest are required if the student

MATHEMATIZING CONSTRUCTIVISM 13

is to make such works objects of his/her experience, if he/she is to achieve them as meaningful in

some manner that connects with his/her life” (36%). This highlights the need to include

collaboration, communication and inquiry into our constructivist math lessons to foster student

development to its greatest potential (A1, Irwin­Gibson, 2016). Therefore, creating context to

provide students with authentic opportunities for construction of knowledge, makes learning real.

MATHEMATIZING CONSTRUCTIVISM 14

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